If the kernel of a 5 x 4 matrix A consists of the zero vector only and if Au = Aw for two vectors v and w in R* then vectors v and w must be equal. True False

Q: M2,2 is the vector space of all 2 x 2 matrices. List the o element of this vector space.

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Q: Find u and usuch that the null space for the matrix U= 3 v= 0 -2 1 1 -3 -15 1 - 9 1 is span{u, }.

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Q: If A and B are n x n matrices then det(A+ B) = det(A) + det(B). true O false

A: If A and B are 2 x 2 matrices such that, A=abcd;B=mnpq Then we get, A+B=abcd+mnpq=a+mb+nc+pd+q det…

Q: 0 5 Let A= 4. diagonalizable matrix( A-XDX), then X and D are: 1 Select one: a. -5 X= -5 0 4 0. b.…

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Q: Construct a matrix that stretches vectors in the direction V₁ = 22-Ĵ by a factor of 4 and leaves…

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Q: > LS A-8 null matrix ?

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Q: Suppose that Q is an orthogonal matrix, then OaQ-8 ObQ-2 d 1Q-0

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Q: If A is an m x n matrix then A' A and A have the same rank. true false

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Q: A is a matrix and there's a vector x, and Ax is defined. Can the equation Ax=0 have no solution ?…

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Q: 19. Can each vector in R4 be written as a linear combination of the columns of the matrix A above?…

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Q: Show that the three points (x1, y1), (x2, y2), and (x3, y3) in a plane are collinear if and only if…

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Q: Suppose A is the 3 by 3 matrix ones (3,3) of all ones. Find two independent vec- tors and y that…

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Q: Show that the matrices [1 1 1] A = |1 3 0 and B = |0 o 0 0 1 1 1 1 are similar.

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Q: Approximate proju and projw. -8 -7 -6 -5 -4 -3 -2 -1 proj u 5.491 projw 4.801 = vector u vector w -5…

A: Given thatWe have to approximate Projwu and Projuw.We know that

Q: To determine whether a set of n vectors from R" is independent, we can form a matrix A whose columns…

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Q: 4. Determine if the vector b is in the span of the columns of the matrix A, where [1 2 3 A = 7 8 10…

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Q: If Q = I- auu' where Qis an orthogonal matrix, a > 0, and u e R"x1 real vector, and I is n x n…

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Q: Suppose A is nxn matrix and the wquation Ax=0 has only the trivial solution. Explain why A has n…

A: If the equation Ax=0 for an n×n matrix A has only a trivial solution, it implies two key things…

Q: Compute(Ax - y)||2, where A = Rm×n is a matrix and x ≤ Rª and y ≤ R™ are both vectors .

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Q: (2) Decide whether the given statement is true of false. Prove your answer. i) If Az = Bx for some…

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Q: Show that if the non-zero vectors s(1), s(2),..., s(m) are conjugate to a positive definite matrix…

A: We are given that the non-zero vectors s(1), s(2), ..., s(n) are conjugate to a positive definite…

Q: Prove that a 2×2 matrix transforms any straight line into a second straight line and points on the…

A: Let the 2×2 matrix be A=a11a12a21a22 which is the matrix of a linear transformation T:ℝ2→ℝ2. Let the…

Q: True/False: Suppose a non-zero n x n matrix A satisfies A³ = 0 (the zero matrix). Then A is not…

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Q: o matrices A and B one has that Null(B) is contained in Null(A · B).

A: Given, for two matrices A and B, to show that Null (B) contained in Null(A·B).

Q: Let S = {(1,0, 1), (0, 1,0)} be two vectors in R³. Write the following vectors as a linear…

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Q: Show that if two real symmetric matrices A, B are congruent then there is a linear change of…

A: The objective of this question is to prove that if two real symmetric matrices A and B are…

Q: If a vector is in the range of a matrix A, then it is also in the null space of A. true of false?

A: Answer

Q: a4 a1 Let V1 (3).. b4 be four vectors in R³. b₁ b₂ V3 = C1 C2 C4 a1 a2 a3 a4 If we reduce the matrix…

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Q: If a 6 x 6 matrix contains 19 zeros, show that there are 4 zeros which are all in distinct rows and…

A: A 6x6 matrix has 36 entries. Given if a 6x6 matrix contains 19 zeros, then to show that there are 4…

Q: rue or false The linear transformation of vectors to Xa = 0 has many solutions if matrix X is less…

A: We know a system of linear equations Ax=b has infinitely many solution if Rank[A:b]=Rank[A]<n…

Q: entries

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Q: Explain why for any matrix A, AA' exists and prove that AA' is a symmetric matrix

A: Let A is any matrix of order m by n Am×n    and   Atn×m Now  AAt exists of order m by m To prove…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.