Suppose A=QR, where Q is m×n and R is n× n. Show that if the columns of A are linearly independent, then R must be invertible. [Hint: Study the equation Rx=0 and use the fact that A=QR.]

Q: Use A-1 =(1/detA)adj A to compute the in­verse of the coefficient matrix 2x + y + 3z = 1     y + z…

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Q: If A ∈ R^n×n is invertible, then columns of A^−1 are linearly independent. Explain why.

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Q: Prove that if the rows of A are linearly dependent, then so are the rows of AB.

A: Let us assume that the product AB makes sense. To prove that is if the rows of A are linearly…

Q: If C is a real square matrix, then C² + I (I is the identity matrix ) is non-singular. - The vector…

A: Find the dimensions of U and V.  Then use the result if two vector spaces have same dimension then…

Q: Is this the correct definition of upper triangular, for a matrix A? Aij * 0 when j 2 i O Yes No

A: Since you have asked multiple questions , we will solve first question for you. If you want any…

Q: Exercise 2.5.14 While trying to invert A, A I is carried to P Q by row operations. Show that P= QA.

A: To show P=QA if AI is carried to PQ by row operations.

Q: Give a matrix representation of the quadratic form f(x, y, z) = 4x 2 + 9y2+ 16z 2 − 2xy + 4xz − 6yz…

A: Given: f(x,y,z)=4x2+9y2+16z2-2xy+4xz-6yz             We have to find a matrix representation of the…

Q: If A is m by n, how many separate multiplications are involved when(a) A multiplies a vector x with…

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Q: Let A be an m×n matrix. Show that if A has linearly independent column vectors, then N(A) = {0}.

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Q: the matrIX A into a product QDQ" where Q is orthogonal and D is diagonal.

A: Given,          A=-5-1-1-1-3-1-1-1-3

Q: Suppose that Q is an orthogonal matrix, then O a lQ°| = 4 O b. Q°| = 2 O d. Q°| = 0

A: This question is related to Linear Algebra

Q: Prove that if two rows (columns) of the n × n matrix A are proportional, then det(A) = 0.

A: Consider the n×n matrix A whose two rows consider R1 and R2 are proportional.…

Q: Find the triangular factorization A = LU for the matrix          2  4  -6  A=    1  5  3         1…

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Q: Given T4 : M2x2 → R² where T )-(p+g. r+s). Find the matrix c) representation of T4 relative to the…

A: Solve.

Q: Let A = ; 1.B - : J.c= , ], anc 2 37 ol.c = [ 1 and v = |1. What is the sum of the entries of the L4…

A: Note: We know that for matrix P of order m*n and Q of order n*t, the multiplication PQ is possible…

Q: For the matrix A in , use the Cayley ­Hamilton Theorem to compute A2, A3, and A4 by expressing each…

A: Given matrix is

Q: Given the set of equations below, 17x+6y+4z+5t=17 8x+22y+2z+4t=20 6x+3y+17z+7t=11 3x+4y+7z+19t=22…

A: We have matrix 1764517822242063177113471922 Cholesky factorization is possible for a symmetric…

Q: Use the concept of area of a parallelogram to write a statement about a 2 × 2 matrix A that is true…

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Q: Find the matrix of the quadratic form. Assume x is in R2. a. 1x21 – 4x1x2 + 5x22 b. 3x21 + 2x1x2

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Q: find the Lineartransfromation T:Maxz (R) /R whose matrix MT with vegard tothe base and the base…

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Q: Find a matrix A such that W = Col A. W [2s-5t] 2t 2s+t :stin R

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Q: Suppose A is (\m \times n \) matrix that has r pivots, then the null space of A contains only the…

A: Option A will be the right answer.

Q: let Q(x)= x12 -8x1x2 -5x22 , then matrix A of the quadratic from is A=?

A: Explanation is given below..

Q: c) [For Susan Foreman of 76 Totter's Lane, Coal Hill School, 1963, WHO Could See This] A a 5x5 real…

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Q: Show that the following are equivalent for an n x n matrix P . (a) P is orthogonal. (b) ||Px|| =…

A: We have to show that the given condition are equivalent for an n×n. First we will show a⇒b Given P…

Q: A" A is invertible if and only if A has linearly independent columns.

A: ATA is invertible if and only if A has linearly independent columns.

Q: Let A be an m × n matrix. Prove that if the reduced echelon form of A possesses precisely one pivot,…

A: Given that A be an m×n matrix. Suppose that reduced echelon form of A possesses precisely one pivot.…

Q: Suppose A is an m x mateix whose columns are linearly independent. What is the nullity of A?

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Q: Solve for the values of x and y using LU Decomposition from the given matrix and prove that PA-LU or…

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Q: (i) Use the properties of trace to argue that tr(AʼA) = tr(AA') for any n × m matrix A. [2 0 (ii)…

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Q: (c) 0 is neither diagonalizable nor

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Q: (d) Suppose A is a 2 × 2 matrix such that for all x, y E R?, (Ax) · (Ay) = x · y Must it be true…

A: This is the problem of linear transformation.

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: Given the polynomial P(x) = x* – 2x° + 2 – 3z + 7, find the remainder of division of P(x) by Q(x) =…

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Q: If an n X n matrix B with det (B) + 0, then for each vector b in R", Bx = b_has several solutions x…

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Q: Let A be an m x n matrix, and let b1,..., bk be vectors in R" such that the system b; has a solution…

A: This question related to linear algebra's topic linearly independent set.

Q: Suppose A is a 2 × 2 matrix, where -2 | u = Au = -4 A?u = 4 A³u 2 20 – 20 – 10. The minimal…

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Q: 5. (a) Show that the columns of the matrix B below span R*. 1 1 0 2 1 3 0 3 0 2 3 3 0 0 6 5 В (3)…

A: See the attachment

Q: Let A be a symmetric 2 X 2 matrix and let k be a scalar. Prove that the graph of the quadratic…

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Q: Let M be a real square matrix such that M3 + M = 2M². Prove that M is diagonalisable.

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Q: Express the quadratic form as v Av, where A is a symmetric matrix. x² + 3y² + 5z² - 4xy+3yz + 10xz

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Q: Prove the following statement: If the columns of A are linearly independent, then AT A is…

A: A is invertible if there exists a matrix A-1 such that AA -1=A -1A=I   The vectors v1,.......,vn are…

Q: Consider the ordered bases B = {1+x, 1 – x} and B' = {u1, u2} for P1(IR), and suppose that - 2 Q =…

A: Given that  β={1+x, 1-x} and β'={u1,u2} are order bases for P1(ℝ) and            Q=2-145    is the…

Q: Show that if the non-zero vectors s1), s(2),..., s) are conjugate to a positive definite matrix G,…

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Q: Use the theorem given below to show that if two rows of a square matrix are equal, then det A = 0.…

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Q: Consider the matrix equation B = P−1AP. Prove that if Ax = x, then PBP−1x = x.

A: The given matrix equation is B=P-1AP.

Q: Let A be a 3 x 3 matrix such that det(A) = 12. Suppose B is obtained from A by applying the…

A: We have to find the  det(B).

Q: Consider the following set of vectors. v, = v, = Let v,, v,, and v, be (column) vectors in R3 and…

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Q: Suppose that a 3 x 3 matrix A factors into a product: 1 0 0111 U12 U13] A = 121 1 0 0 U22 U23 Ll31…

A: Given that  A = 100l2110l31l321 u11u12u130u22u2300u33 We have to determine the value of  det A.

Q: For A e Mnxn(C), the function ||A||. = max1<ijśn |aij| is a norm on Mnxn(C) but not a matrix norm.…

A: Matrix Norm: Let Mn×n(ℂ) be the vector space of all matrices of order n with entries in the field ℂ,…

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.