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EBK DIFFERENTIAL EQUATIONS AND LINEAR A
- Consider an mn matrix A and an np matrix B. Show that the row vectors of AB are in the row space of B and the column vectors of AB are in the column space of A.arrow_forwardCan a matrix with zeros on the diagonal have an inverse? If so, find an example. If not, prove why not. For simplicity, assume a 22 matrix.arrow_forwardDetermine if the statement is true or false. If the statement is false, then correct it and make it true. For the product of two matrices to be defined, the number of rows of the first matrix must equal the number of columns of the second matrix.arrow_forward
- Get a base for the column space of the matrix shown in the imagearrow_forwardLet A be an n x n matrix, let S = (ü, ü, u) be a set of non-zero vectors in R", and let U be a subspace of R" of dimension at least 1. Look at the expressions and phrases that follow. Select the ones that DO NOT make sense, because they either equate two different "types" of thing that can't be equal, use nonsensical notation, or try to perform an operation that is not defined. (For the computer programmers reading this, the question is essentially "find the type errors".) Warning: To clarify, you're not being asked which ones are true. You're being asked to identify which equations don't make sense. From the 11 choices, select all that apply "a basis of 5" "the solutions of the system of equations" im(A) (R" Ay for some FR") "the span of A" im(A) - (A#-5) "S spans U "the solutions of the matrix" im(A) {ER: Af=ÿ) null(A) = {A=0} 1 null(4)-(ER": Až=6) "U spans S"arrow_forwardSuppose A is a 3x3 matrix. Let B be the matrix obtained from A by performing the following elementary row operations on A: Firstly, R, + R, then replace R, with 4R, + R, . Express B as a product of A and elementa matrices.arrow_forward
- 3. (a) Let A = (b) Let A = 1 1 0 1 1 1 0 1 1 1 1 combination of the column vectors of A, with coefficeints x1, x2, and x3. (c) Now generalize part (b) to the case where A is m × n. That is, show that if A is an 1 1 0 1 1 1 1 0 1 1 1 18 - What are the row vectors and column vectors of this matrix? 9 arbitrary mx n matrix and x = m and let = X1 x2 x 3 X1 x2 In the columns of A, with coefficients x1, x2,., n. be a vector in R³. Show that A is a linear 2c1 c₂ + c3, and let (d) Suppose that A is a 4 × 3 matrix, and let c₁, c2, c3 be the column vectors of A. That A [C₁ C₂ C3]. Suppose that be R4 is the vector 6 = 20₁ [ci is, is in R", then A is a linear combination of 9 X1 = X2 . Show that Aỡ = b is consistent by finding a solution to the system.arrow_forwardThe set of non-zero vectors {b,, b,, bz, b4} are vectors in R* with the property that 4b, + 2b, = 6b, . Let matrix B = | b, b2 b; b4. (a) What does the matrix b, b, b, reduce to in RREF form? (b) Find a specific solution to the matrix equation Bx = 0. That is, find an x vector that solves that equation. (c) What is the maximum number of pivots matrix B can have? Please explain briefly referencing anything from above. (d) Can the set of vectors {b1, b2, b3, b4} span R’? Explain why or why not? (e) Can the set of vectors {b,, b2, b3, b4} span R*? Explain why or why not? (f) Matrix M is 7 x 4. If possible, show that the columns of the matrix MB are linearly dependent. If it is not possible to show this, explain why.arrow_forward
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