Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN: 9781305658004
Author: Ron Larson
Publisher: Cengage Learning
bartleby

Videos

Question
Book Icon
Chapter 7.CM, Problem 1CM
To determine

To prove:

If T:R3R2, T(x,y,z)=(2x,x+y) is a linear transformation.

Expert Solution & Answer
Check Mark

Answer to Problem 1CM

Solution:

The given function T is a linear transformation.

Explanation of Solution

Given:

The function T:R3R2, T(x,y,z)=(2x,x+y).

Approach:

A function T:AB is a linear transformation if for any two vectors, a,bA and any scalar c, the following two properties hold:

1) T(a+b)=T(a)+T(b)

2) T(ca)=cT(a)

Calculation:

Consider any two elements, a,bR3.

Then, a and b are of the form:

a=(a1,a2,a3)b=(b1,b2,b3)

Consider,

T(a+b)=T(a1+b1,a2+b2,a3+b3)=(2(a1+b1),a1+b1+a2+b2)=(2a1,a1+a2)+(2b1,b1+b2)=T(a)+T(b)

So, property 1 is true for T.

Now, consider,

T(ca)=T(ca1,ca2,ca3)=(2(ca1),ca1+ca2)=c(2a1,a1+a2)=cT(a)

So, property 2 is true for T.

Since, both the properties of a linear transformation are satisfied by T, it is a linear transformation.

Therefore, the given function T is a linear transformation.

Conclusion:

Hence, the given function T is a linear transformation.

Want to see more full solutions like this?

Subscribe now to access step-by-step solutions to millions of textbook problems written by subject matter experts!
Students have asked these similar questions
Assume that you fancy polynomial splines, while you actually need ƒ(t) = e²/3 – 1 for t€ [−1, 1]. See the figure for a plot of f(t). Your goal is to approximate f(t) with an inter- polating polynomial spline of degree d that is given as sa(t) = • Σk=0 Pd,k bd,k(t) so that sd(tk) = = Pd,k for tk = −1 + 2 (given d > 0) with basis functions bd,k(t) = Σi±0 Cd,k,i = • The special case of d 0 is trivial: the only basis function b0,0 (t) is constant 1 and so(t) is thus constant po,0 for all t = [−1, 1]. ...9 The d+1 basis functions bd,k (t) form a ba- sis Bd {ba,o(t), ba,1(t), bd,d(t)} of the function space of all possible sα (t) functions. Clearly, you wish to find out, which of them given a particular maximal degree d is the best-possible approximation of f(t) in the least- squares sense. _ 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 function f(t) = exp((2t)/3) - 1 to project -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5…
An image processor considered a 750×750 pixels large subset of an image and converted it into gray-scale, resulting in matrix gIn - a false-color visualization of gIn is shown in the top-left below. He prepared a two-dim. box filter f1 as a 25×25 matrix with only the 5×5 values in the middle being non-zero – this filter is shown in the top-middle position below. He then convolved £1 with itself to get £2, before convolving £2 with itself to get f3. In both of the steps, he maintained the 25×25 size. Next, he convolved gIn with £3 to get gl. Which of the six panels below shows g1? Argue by explaining all the steps, so far: What did the image processor do when preparing ₤3? What image processing operation (from gin to g1) did he prepare and what's the effect that can be seen? Next, he convolved the rows of f3 with filter 1/2 (-1, 8, 0, -8, 1) to get f4 - you find a visualization of filter f 4 below. He then convolved gIn with f4 to get g2 and you can find the result shown below. What…
3ur Colors are enchanting and elusive. A multitude of color systems has been proposed over a three-digits number of years - maybe more than the number of purposes that they serve... - Everyone knows the additive RGB color system – we usually serve light-emitting IT components like monitors with colors in that system. Here, we use c = (r, g, b) RGB with r, g, bЄ [0,1] to describe a color c. = T For printing, however, we usually use the subtractive CMY color system. The same color c becomes c = (c, m, y) CMY (1-c, 1-m, 1-y) RGB Note how we use subscripts to indicate with coordinate system the coordinates correspond to. Explain, why it is not possible to find a linear transformation between RGB and CMY coordinates. Farbenlehr c von Goethe Erster Band. Roſt einen Defte mit fergen up Tübingen, is et 3. Cotta'fden Babarblung. ISIO Homogeneous coordinates give us a work-around: If we specify colors in 4D, instead, with the 4th coordinate being the homogeneous coordinate h so that every actual…

Chapter 7 Solutions

Elementary Linear Algebra (MindTap Course List)

Ch. 7.1 - Determining Eigenvectors In Exercise 9-12,...Ch. 7.1 - Prob. 12ECh. 7.1 - Prob. 13ECh. 7.1 - Prob. 14ECh. 7.1 - Characteristic Equation, Eigenvalues, and...Ch. 7.1 - Characteristic Equation, Eigenvalues, and...Ch. 7.1 - Characteristic Equation, Eigenvalues, and...Ch. 7.1 - Prob. 18ECh. 7.1 - Characteristic Equation, Eigenvalues, and...Ch. 7.1 - Prob. 20ECh. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Prob. 24ECh. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Characteristic Equation, Eigenvalues and...Ch. 7.1 - Prob. 29ECh. 7.1 - Prob. 30ECh. 7.1 - Prob. 31ECh. 7.1 - Prob. 32ECh. 7.1 - Prob. 33ECh. 7.1 - Prob. 34ECh. 7.1 - Prob. 35ECh. 7.1 - Prob. 36ECh. 7.1 - Prob. 37ECh. 7.1 - Prob. 38ECh. 7.1 - Prob. 39ECh. 7.1 - Finding EigenvaluesIn Exercises 29-40, use a...Ch. 7.1 - Eigenvalues of Triangular and Diagonal Matrices In...Ch. 7.1 - Eigenvalues of Triangular and Diagonal Matrices In...Ch. 7.1 - Prob. 43ECh. 7.1 - Eigenvalues of Triangular and Diagonal Matrices In...Ch. 7.1 - Eigenvalues and Eigenvectors of Linear...Ch. 7.1 - Prob. 46ECh. 7.1 - Eigenvalues and Eigenvectors of Linear...Ch. 7.1 - Eigenvalues and Eigenvectors of Linear...Ch. 7.1 - Cayley-Hamilton TheoremIn Exercises 49-52,...Ch. 7.1 - Cayley-Hamilton TheoremIn Exercises 49-52,...Ch. 7.1 - Prob. 51ECh. 7.1 - Prob. 52ECh. 7.1 - Prob. 53ECh. 7.1 - Prob. 54ECh. 7.1 - Prob. 55ECh. 7.1 - Prob. 56ECh. 7.1 - Prob. 57ECh. 7.1 - Proof Prove that A and AT have the same...Ch. 7.1 - Prob. 59ECh. 7.1 - Define T:R2R2 by T(v)=projuv Where u is a fixed...Ch. 7.1 - Prob. 61ECh. 7.1 - Prob. 62ECh. 7.1 - Prob. 63ECh. 7.1 - Prob. 64ECh. 7.1 - Prob. 65ECh. 7.1 - Show that A=[0110] has no real eigenvalues.Ch. 7.1 - True or False? In Exercises 67 and 68, determine...Ch. 7.1 - True or False? In Exercises 67 and 68, determine...Ch. 7.1 - Finding the Dimension of an Eigenspace In...Ch. 7.1 - Finding the Dimension of an Eigenspace In...Ch. 7.1 - Prob. 71ECh. 7.1 - Prob. 72ECh. 7.1 - Prob. 73ECh. 7.1 - Prob. 74ECh. 7.1 - Prob. 75ECh. 7.1 - Define T:P2P2 by...Ch. 7.1 - Prob. 77ECh. 7.1 - Find all values of the angle for which the matrix...Ch. 7.1 - Prob. 79ECh. 7.1 - Prob. 80ECh. 7.1 - Prob. 81ECh. 7.2 - Diagonalizable Matrices and Eigenvalues In...Ch. 7.2 - Diagonalizable Matrices and Eigenvalues In...Ch. 7.2 - Diagonalizable Matrices and Eigenvalues In...Ch. 7.2 - Diagonalizable Matrices and Eigenvalues In...Ch. 7.2 - Diagonalizable Matrices and Eigenvalues In...Ch. 7.2 - Prob. 6ECh. 7.2 - Diagonalizing a Matrix In Exercise 7-14, find if...Ch. 7.2 - Prob. 8ECh. 7.2 - Diagonalizing a Matrix In Exercise 7-14, find if...Ch. 7.2 - Diagonalizing a Matrix In Exercise 7-14, find if...Ch. 7.2 - Diagonalizing a Matrix In Exercise 7-14, find if...Ch. 7.2 - Diagonalizing a Matrix In Exercise 7-14, find if...Ch. 7.2 - Diagonalizing a Matrix In Exercise 7-14, find if...Ch. 7.2 - Prob. 14ECh. 7.2 - Show That a Matrix Is Not Diagonalizable In...Ch. 7.2 - Prob. 16ECh. 7.2 - Show That a Matrix Is Not Diagonalizable In...Ch. 7.2 - Show That a Matrix Is Not Diagonalizable In...Ch. 7.2 - Show That a Matrix Is Not Diagonalizable In...Ch. 7.2 - Show That a Matrix Is Not Diagonalizable In...Ch. 7.2 - Prob. 21ECh. 7.2 - Prob. 22ECh. 7.2 - Determine a Sufficient Condition for...Ch. 7.2 - Determine a Sufficient Condition for...Ch. 7.2 - Determine a Sufficient Condition for...Ch. 7.2 - Determine a Sufficient Condition for...Ch. 7.2 - Finding a Basis In Exercises 27-30, find a basis B...Ch. 7.2 - Finding a Basis In Exercises 27-30, find a basis B...Ch. 7.2 - Prob. 29ECh. 7.2 - Prob. 30ECh. 7.2 - Prob. 31ECh. 7.2 - Prob. 32ECh. 7.2 - Prob. 33ECh. 7.2 - Finding a Power of a Matrix In Exercises 33-36,...Ch. 7.2 - Prob. 35ECh. 7.2 - Prob. 36ECh. 7.2 - True or False? In Exercises 37 and 38, determine...Ch. 7.2 - True or False? In Exercises 37 and 38, determine...Ch. 7.2 - Are the two matrices similar? If so, find a matrix...Ch. 7.2 - Prob. 40ECh. 7.2 - Prob. 41ECh. 7.2 - Proof Prove that if matrix A is diagonalizable,...Ch. 7.2 - Proof Prove that if matrix A is diagonalizable...Ch. 7.2 - Prob. 44ECh. 7.2 - Prob. 45ECh. 7.2 - Guide Proof Prove nonzero nilpotent matrices are...Ch. 7.2 - Prob. 47ECh. 7.2 - CAPSTONE Explain how to determine whether an nn...Ch. 7.2 - Prob. 49ECh. 7.2 - Showing That a Matrix Is Not Diagonalizable In...Ch. 7.3 - Determining Whether a Matrix Is Symmetric In...Ch. 7.3 - Prob. 2ECh. 7.3 - Proof In Exercise 3-6, prove that the symmetric...Ch. 7.3 - Prob. 4ECh. 7.3 - Prob. 5ECh. 7.3 - Prob. 6ECh. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Finding Eigenvalues and Dimensions of Eigen spaces...Ch. 7.3 - Prob. 14ECh. 7.3 - Prob. 15ECh. 7.3 - Prob. 16ECh. 7.3 - Prob. 17ECh. 7.3 - Prob. 18ECh. 7.3 - Determine Whether a Matrix Is Orthogonal In...Ch. 7.3 - Prob. 20ECh. 7.3 - Prob. 21ECh. 7.3 - Prob. 22ECh. 7.3 - Prob. 23ECh. 7.3 - Prob. 24ECh. 7.3 - Prob. 25ECh. 7.3 - Prob. 26ECh. 7.3 - Prob. 27ECh. 7.3 - Prob. 28ECh. 7.3 - Prob. 29ECh. 7.3 - Prob. 30ECh. 7.3 - Prob. 31ECh. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Prob. 34ECh. 7.3 - Prob. 35ECh. 7.3 - Eigenvectors of Symmetric Matrix In Exercises...Ch. 7.3 - Prob. 37ECh. 7.3 - Prob. 38ECh. 7.3 - Prob. 39ECh. 7.3 - Orthogonally Diagonalizable Matrices In Exercise...Ch. 7.3 - Prob. 41ECh. 7.3 - Prob. 42ECh. 7.3 - Prob. 43ECh. 7.3 - Prob. 44ECh. 7.3 - Orthogonal Diagonalization In Exercise 43-52, find...Ch. 7.3 - Orthogonal Diagonalization In Exercise 43-52, find...Ch. 7.3 - Orthogonal Diagonalization In Exercise 4-52, find...Ch. 7.3 - Prob. 48ECh. 7.3 - Prob. 49ECh. 7.3 - Orthogonal Diagonalization In Exercise 43-52, find...Ch. 7.3 - Orthogonal Diagonalization In Exercise 4-52, find...Ch. 7.3 - Prob. 52ECh. 7.3 - Prob. 53ECh. 7.3 - Prob. 54ECh. 7.3 - Prob. 55ECh. 7.3 - Prob. 56ECh. 7.3 - Prob. 57ECh. 7.3 - Prob. 58ECh. 7.3 - Prob. 59ECh. 7.3 - Find ATA and AAT for the matrix below. What do you...Ch. 7.4 - Finding Age Distribution Vectors In Exercises 1-6,...Ch. 7.4 - Prob. 2ECh. 7.4 - Prob. 3ECh. 7.4 - Finding Age Distribution Vectors In Exercises 1-6,...Ch. 7.4 - Prob. 5ECh. 7.4 - Prob. 6ECh. 7.4 - Population Growth Model A population has the...Ch. 7.4 - Population Growth Model A population has the...Ch. 7.4 - Prob. 9ECh. 7.4 - Find the limit if it exists of Anx1 as n...Ch. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - Prob. 13ECh. 7.4 - Prob. 14ECh. 7.4 - Prob. 15ECh. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Prob. 20ECh. 7.4 - Solving a System of Linear Differential Equations...Ch. 7.4 - Solving a System of Linear Differential Equations...Ch. 7.4 - Prob. 23ECh. 7.4 - Solving a System of Linear Differential Equations...Ch. 7.4 - Prob. 25ECh. 7.4 - Prob. 26ECh. 7.4 - Solving a System of Linear Differential Equations...Ch. 7.4 - Prob. 28ECh. 7.4 - Prob. 29ECh. 7.4 - Prob. 30ECh. 7.4 - Prob. 31ECh. 7.4 - Prob. 32ECh. 7.4 - Prob. 33ECh. 7.4 - Prob. 34ECh. 7.4 - Prob. 35ECh. 7.4 - Prob. 36ECh. 7.4 - Prob. 37ECh. 7.4 - Prob. 38ECh. 7.4 - Prob. 39ECh. 7.4 - Prob. 40ECh. 7.4 - Prob. 41ECh. 7.4 - Prob. 42ECh. 7.4 - Prob. 43ECh. 7.4 - Prob. 44ECh. 7.4 - Prob. 45ECh. 7.4 - Prob. 46ECh. 7.4 - Rotation of a Conic In Exercises 45-52, use the...Ch. 7.4 - Prob. 48ECh. 7.4 - Prob. 49ECh. 7.4 - Prob. 50ECh. 7.4 - Prob. 51ECh. 7.4 - Prob. 52ECh. 7.4 - Prob. 53ECh. 7.4 - Prob. 54ECh. 7.4 - Prob. 55ECh. 7.4 - Prob. 56ECh. 7.4 - Prob. 57ECh. 7.4 - Prob. 58ECh. 7.4 - Prob. 59ECh. 7.4 - Prob. 60ECh. 7.4 - Prob. 61ECh. 7.4 - Prob. 62ECh. 7.4 - Prob. 63ECh. 7.4 - Prob. 64ECh. 7.4 - Prob. 65ECh. 7.4 - Prob. 66ECh. 7.4 - Prob. 67ECh. 7.4 - Use your schools library, the Internet, or some...Ch. 7.CR - Characteristic Equation, Eigenvalues, and Basis In...Ch. 7.CR - Characteristic Equation, Eigenvalues, and Basis In...Ch. 7.CR - Characteristic Equation, Eigenvalues, and Basis In...Ch. 7.CR - Prob. 4CRCh. 7.CR - Characteristic Equation, Eigenvalues, and Basis In...Ch. 7.CR - Prob. 6CRCh. 7.CR - Characteristics Equation, Eigenvalues, and Basis...Ch. 7.CR - Characteristics Equation, Eigenvalues, and Basis...Ch. 7.CR - Determining Whether a Matrix Is DiagonalizableIn...Ch. 7.CR - Prob. 10CRCh. 7.CR - Determining Whether a Matrix Is DiagonalizableIn...Ch. 7.CR - Prob. 12CRCh. 7.CR - Determining Whether a Matrix Is DiagonalizableIn...Ch. 7.CR - Prob. 14CRCh. 7.CR - For what values of a does the matrix A=[01a1] have...Ch. 7.CR - Prob. 16CRCh. 7.CR - Writing In Exercises 17-20, explain why the given...Ch. 7.CR - Prob. 18CRCh. 7.CR - Writing In Exercises 17-20, explain why the given...Ch. 7.CR - Prob. 20CRCh. 7.CR - Determine Whether Two Matrices Are Similar In...Ch. 7.CR - Determine Whether Two Matrices Are Similar In...Ch. 7.CR - Determine Whether Two Matrices Are Similar In...Ch. 7.CR - Determine Whether Two Matrices Are Similar In...Ch. 7.CR - Determining Symmetric and Orthogonal Matrices In...Ch. 7.CR - Prob. 26CRCh. 7.CR - Determining Symmetric and Orthogonal Matrices In...Ch. 7.CR - Prob. 28CRCh. 7.CR - Prob. 29CRCh. 7.CR - Determine Symmetric and Orthogonal Matrices In...Ch. 7.CR - Prob. 31CRCh. 7.CR - Prob. 32CRCh. 7.CR - Prob. 33CRCh. 7.CR - Prob. 34CRCh. 7.CR - Prob. 35CRCh. 7.CR - Prob. 36CRCh. 7.CR - Orthogonally Diagonalizable Matrices In Exercises...Ch. 7.CR - Prob. 38CRCh. 7.CR - Orthogonally Diagonalizable Matrices In Exercises...Ch. 7.CR - Prob. 40CRCh. 7.CR - Prob. 41CRCh. 7.CR - Prob. 42CRCh. 7.CR - Prob. 43CRCh. 7.CR - Prob. 44CRCh. 7.CR - Prob. 45CRCh. 7.CR - Orthogonal Diagonalization In Exercises 41-46,...Ch. 7.CR - Prob. 47CRCh. 7.CR - Prob. 48CRCh. 7.CR - Prob. 49CRCh. 7.CR - Prob. 50CRCh. 7.CR - Prob. 51CRCh. 7.CR - Prob. 52CRCh. 7.CR - Steady State Probability Vector In Exercises...Ch. 7.CR - Prob. 54CRCh. 7.CR - Prob. 55CRCh. 7.CR - Prob. 56CRCh. 7.CR - Prob. 57CRCh. 7.CR - Prob. 58CRCh. 7.CR - Prob. 59CRCh. 7.CR - Prob. 60CRCh. 7.CR - Prob. 61CRCh. 7.CR - Prob. 62CRCh. 7.CR - Prob. 63CRCh. 7.CR - a Find a symmetric matrix B such that B2=A for...Ch. 7.CR - Determine all nn symmetric matrices that have 0 as...Ch. 7.CR - Prob. 66CRCh. 7.CR - Prob. 67CRCh. 7.CR - Prob. 68CRCh. 7.CR - Prob. 69CRCh. 7.CR - True or False? In Exercises 69 and 70, determine...Ch. 7.CR - Prob. 71CRCh. 7.CR - Prob. 72CRCh. 7.CR - Prob. 73CRCh. 7.CR - Prob. 74CRCh. 7.CR - Prob. 75CRCh. 7.CR - Prob. 76CRCh. 7.CR - Prob. 77CRCh. 7.CR - Prob. 78CRCh. 7.CR - Prob. 79CRCh. 7.CR - Prob. 80CRCh. 7.CR - Prob. 81CRCh. 7.CR - Prob. 82CRCh. 7.CR - Prob. 83CRCh. 7.CR - Prob. 84CRCh. 7.CR - Prob. 85CRCh. 7.CR - Prob. 86CRCh. 7.CR - Prob. 87CRCh. 7.CR - Prob. 88CRCh. 7.CM - Prob. 1CMCh. 7.CM - In Exercises 1 and 2, determine whether the...Ch. 7.CM - Let T:RnRm be the linear transformation defined by...Ch. 7.CM - Prob. 4CMCh. 7.CM - Find the kernel of the linear transformation...Ch. 7.CM - Let T:R4R2 be the linear transformation defined by...Ch. 7.CM - In Exercises 7-10, find the standard matrix for...Ch. 7.CM - Prob. 8CMCh. 7.CM - Prob. 9CMCh. 7.CM - Prob. 10CMCh. 7.CM - Prob. 11CMCh. 7.CM - Prob. 12CMCh. 7.CM - Prob. 13CMCh. 7.CM - Prob. 14CMCh. 7.CM - Prob. 15CMCh. 7.CM - Prob. 16CMCh. 7.CM - Prob. 17CMCh. 7.CM - Prob. 18CMCh. 7.CM - In Exercises 19-22, find the eigenvalues and the...Ch. 7.CM - Prob. 20CMCh. 7.CM - Prob. 21CMCh. 7.CM - Prob. 22CMCh. 7.CM - In Exercises 23 and 24, find a nonsingular matrix...Ch. 7.CM - In Exercises 23 and 24, find a nonsingular matrix...Ch. 7.CM - Find a basis B for R3 such that the matrix for the...Ch. 7.CM - Find an orthogonal matrix P such that PTAP...Ch. 7.CM - Use the Gram-Schmidt orthonormalization process to...Ch. 7.CM - Prob. 28CMCh. 7.CM - Prob. 29CMCh. 7.CM - Prob. 30CMCh. 7.CM - Prob. 31CMCh. 7.CM - Prove that if A is similar to B and A is...
Knowledge Booster
Background pattern image
Algebra
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.
Similar questions
SEE MORE QUESTIONS
Recommended textbooks for you
Text book image
Elementary Linear Algebra (MindTap Course List)
Algebra
ISBN:9781305658004
Author:Ron Larson
Publisher:Cengage Learning
Text book image
Linear Algebra: A Modern Introduction
Algebra
ISBN:9781285463247
Author:David Poole
Publisher:Cengage Learning
Text book image
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:9781133382119
Author:Swokowski
Publisher:Cengage
Text book image
Big Ideas Math A Bridge To Success Algebra 1: Stu...
Algebra
ISBN:9781680331141
Author:HOUGHTON MIFFLIN HARCOURT
Publisher:Houghton Mifflin Harcourt
Linear Transformations on Vector Spaces; Author: Professor Dave Explains;https://www.youtube.com/watch?v=is1cg5yhdds;License: Standard YouTube License, CC-BY
Linear Equation | Solving Linear Equations | What is Linear Equation in one variable ?; Author: Najam Academy;https://www.youtube.com/watch?v=tHm3X_Ta_iE;License: Standard YouTube License, CC-BY