Linear algebra Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by \( T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = a + b + c + dx^2 \).### A) A basis for the image (range) of \( T \) would be:- \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\)- \(\{1, 1, 1, x^2\}\)- \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\)- \(\{0\}\)- \(\{1, x, x^2\}\)- **\(\{1, x^2\}\)** (Correct Answer)- \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{Bmatrix}\)### B) A basis for the kernel of \( T \) would be:- \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\)- \(\{0\}\)- \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{Bmatrix}\)- \(\{1, x^2\}\)- \(\{1, 1, 1, x^2\}\)- **\(\{1, x, x^2\}\)** (Correct Answer)- \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\)

Answered: Image /qna-images/answer/49c63620-1c5f-4cca-ac81-00aef869d865.jpg

Let T: M2x2 → P₂ be defined by T A) A basis for the image (range) of T would be: {[][]} O{1, 1, 1, 2²} {[33]} o{0} O{1, x, x²} Ⓒ{1, 2²} {[89]} B) A basis for the kernel of T would be: {[3]} o {0} {[89]} O{1, ²} O{1, 1, 1, ²} {} =a+b+c+dx².

Let T: M2x2 → P₂ be defined by T A) A basis for the image (range) of T would be: {[][]} O{1, 1, 1, 2²} {[33]} o{0} O{1, x, x²} Ⓒ{1, 2²} {[89]} B) A basis for the kernel of T would be: {[3]} o {0} {[89]} O{1, ²} O{1, 1, 1, ²} {} =a+b+c+dx².

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

See similar textbooks

Related questions

Q: Determine whether v is in the column space of A, or the null space of A, or both, or neither. -8 -2…

A: First, perform the row reduction to get the row reduction echelon form. Solve Ax=0 to get the…

Q: Aigebia 20 qucricca und Ociie3 All chang 9. An Infinite geometric series has a first term aj = 15…

Q: Which statement regarding the regression is false? the dependent variable is assignment to the…

A: We have to answer question on regression.

Q: Let A > 0 be such that each row (or column) sum of A equals r. Show that p(A) = r

A: Let A>0 be a matrix such that each row or column sum of A is equal to r.Show that:

Q: speed (km / h) 52 54 54 71 42 51 51 67 54 po 46 54 63 55 braking distance 69 75 74 126 48 69 66 115…

A: Given data is Here, x denotes speed (km/h) y denotes braking distance (m) We need to find the…

Q: EXER 3.1 Find a symmetric matrix E (E = ET) and a skew-symmetric Justify all parts of the solution.…

Q: Describe the procedures to find the factor of a matrix, matrix norm, column norm and row norm.

A: Factor of a matrix: Let, A=1abc1bac1cab Performing row operations, 1abc1bac1cab…

Q: Product of matrices multiplication A=[a, b; c, d] x[1,0;0,1]

A: In the question we have to find the product of the matrix.

Q: Using the same data as problem 2 above x y ху 2 7 4 11 5 16 8 12 10 18 Then the value of [ Select ]…

A: Formulae : i ) x¯ = ∑xn ii) y¯ = ∑yn iii) Slope ( b1 ) : iv) Y-intercept ( b0 ) :

Q: Which line has the least rate of change? A. The line passing through (2,4) and (5,13) B. The line…

A: By using rate of change formula, we calculate the least rate of change.

Q: (a) Calculate (a +36). C where a = (0, -1, 0, 1), 6 = (1,-1, 1, -1) and c = (2,2,3, -3) (b)…

Q: 1. Write a linear equation using matrices that passes through the points (-2, -1), (2,5) and (4, 8)

A: Aim: To find the linear equation of line using matrices. The line is passing through -2,-1, 2,5,…

Q: (1 2 -3 1 -3 0 3 2 4X=| 10 2 7 2 -1 0 10 7 8

Q: Explain Matrix-Valued Functions?

A: A function is a relation from a set A to set B such that each values in the set A maps to exactly…

Q: Linear Quadratic Gaussian Control use ?

A: Linear Quadratic Gaussian Control use :

Q: understand

A: We will simplify the expressions.

Q: 3 Find x such that the matrix B = is -2 -3 equal to its own inverse. а. 4 с. 9 b. 6 d. 10

Q: Q1) For primary producers such as plants, photosynthesis requires carbon dioxide (CO₂), water (H₂O),…

A: Photosynthesis is a process by which plants make glucose and release oxygen with the help of…

Q: Explain The Chain Rule?

A: The Chain rule is applicable for the differentiating the composite functions. First derivate the…

Q: A set of measurements X has a mean of 7 and standard deviation of 0.2 for simplicity, a linear…

A: Mean of X = 7 Standard deviation of X = 0.2 Linear transformation Y = a X + b Mean of Y = 1…

Q: 17. The inverse of matrix a. Matrix 11 Matrix 1 1/2 | =11 b. 0 c. Matrix ₁ d. Matrix ¹

A: Given matrix is A = 1101

Q: How do you do multivariate linear regression model; and solve for y hat training, y hat testing and…

A: Multivariate linear regression : This is quite like the easy regression model it's mentioned…

Q: Find the inverse of M 1 3] 1 2 2] 2 M : 1 Lo 1

Q: Find the least squares regression line.(−3, 4), (−1, 2), (1, 1), (3, 0)

Q: Consider the matrix A and the reduced row echelon form of A. 1 2 1 41[1 0 -1 A = 4 5 1 7 1 1 6 7 1 8…

A: We will use the definition.

Q: Find the least squares regression line, using the matrix form for linear regression. (−7, 1), (1,…

A: The linear relationship between two variables x and y can be defined using the equation, y=b+mx. The…

Q: row echelon form

A: in row echelon form we use row operations to convert given matrix in upper triangular type matrix.…

Q: Algebra

A: In order to find the eigen values of T and associated eigen space of T i.e T* First we find T* Then…

Q: Chi square is used to test __________________ dependent variables.

A: Solution: Nominal

Q: Write v as a linear combination of u and w, if possible, where u = (3, 1) and w=(3,-2). (Enter your…

A: In mathematics, a linear combination is a mathematical expression or operation that combines…

Q: Using linear interpolation, which value of Reactor Temperature is closest to the Reactor Temperature…

A: Consider the given table, The linear interpolation formula is defined as,

Q: Count of Income Educ. Income 2. 4 Grand Total 55 169 141 426 355 61 33 125 122 75 39 175 3. 33 43…

A: Probability indicates the chances of occurring an event. It lies between 0 and 1. zero probability…

Q: ving matrix op

Q: Linear Algebra Row Reduce. Is this right? I think it is wrong because the third row should be 0 0 0…

Question

Linear algebra

Let \( T: M_{2 \times 2} \rightarrow P_2 \) be defined by \( T \left( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \right) = a + b + c + dx^2 \).

### A) A basis for the image (range) of \( T \) would be:
- \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 1 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\)
- \(\{1, 1, 1, x^2\}\)
- \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\)
- \(\{0\}\)
- \(\{1, x, x^2\}\)
- **\(\{1, x^2\}\)** (Correct Answer)
- \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 1 \end{bmatrix} \end{Bmatrix}\)

### B) A basis for the kernel of \( T \) would be:
- \(\begin{Bmatrix} \begin{bmatrix} 2 \\ -1 \\ 0 \end{bmatrix} \end{Bmatrix}\)
- \(\{0\}\)
- \(\begin{Bmatrix} \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \end{Bmatrix}\)
- \(\{1, x^2\}\)
- \(\{1, 1, 1, x^2\}\)
- **\(\{1, x, x^2\}\)** (Correct Answer)
- \(\begin{Bmatrix} \begin{bmatrix} 1 & -1 \\ 1 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0 \\ -1 & 0 \end{bmatrix} \end{Bmatrix}\)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

SEE SOLUTION Check out a sample Q&A here

Step 1: Introduction of the given problem

VIEW

Step 2: Finding the basis of the image of T

VIEW

Step 3: Finding the Kernel of T

VIEW

Solution

VIEW

Step by step

Solved in 4 steps with 14 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

Linear Algebra
A my Question 1 Consider each of the following scatterplots, as made on the TI. This relationship is: negative linear no relationship non-linear relationship O positive linear O O
Pls Help ASAP
Interpretation of the estimated coefficients of a Linear Regression Equation
MAT280 Fall2020 TakeHome 119 update - Word Riley Cyron RC Mailings Review View Help A Share Home Insert Design Layout References Let å = (first two ID digits), = (middle three ID digits), 2=| 1 (last two digits) Let i = (first three digits), v = (last four digits) 1 5 7 4 9 Let A = (place your whole ID in each row) 15 7 4 9 1 Calculations. 1. Compute (u, 3). Determine ||v||. Write the answer as a simplified square root. 3. Calculate the angle between u and v in radians. Round answer to four decimal places. 4. Construct projzv. Round each component to four decimal places. 5. Let p = 2+3x - x3. Determine the coordinate matrix of p relative to the basis B = {1,1 – x, 1 + x²,1 – x3}. 6. Construct the orthogonal complement S for S = span{u, v}. 7. Use the Gram-Schmidt Process to construct an orthonormal basis for the vector space V = span{d, b, č}. Round answers to four decimal places, where necessary. 8. Consider the basis B = {(2, -V2), (2, V2)} for R?. Define (3, z) =y,z1 +y22. %3D a)…
Linear Algebra