2."!(a)(b)Given the matrices A and the reduced echelon form of A below, find:i. a basis for Row(A). What is the dimension of Row(A)?ii. a basis for Col(A). What is the dimension of Col(A)?iii. a basis for Nul(A). What is the dimension of Nul(A)?A1234 51012 -12246 411 20 1Consider vectorsrref (A) =0 1 001 1 00 0 0 10000-5/38/31/30V₁ = [1,-1, 5, 2], V₂ = [-2, 3, 1, 0], V3 = [9, -10, 34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8]in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}.10010000NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is705-1 0 4016000marks) Let A be an m x n matrix. Prove that every vector in the null space of A isorthogonal to every vector in the row space of A.

We have to given matrix A and his row reduced echelon form of A.

(a) i. a basis for Row(A). What is the dimension of Row(A)? ii. a basis for Col(A). What is the dimension of Col(A)? iii. a basis for Nul(A). What is the dimension of Nul(A)? Given the matrices A and the reduced echelon form of A below, find: (b) A = 1234 5 1 0 1 2 -1 2246 4 1120 1 Consider vectors rref (A) = 1 0 1 0 0 1 1 0 0 0 0 1 00000 -5/3 8/3 1/3 V₁ = [1,-1, 5, 2], V2 = [−2, 3, 1, 0], V3 = [9, -10,34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8] in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}. 10 705 0 1 -1 04 00 016 00 000 NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is marks) Let A be an m x n matrix. Prove that every vector in the null space of A is orthogonal to every vector in the row space of A.

(a) i. a basis for Row(A). What is the dimension of Row(A)? ii. a basis for Col(A). What is the dimension of Col(A)? iii. a basis for Nul(A). What is the dimension of Nul(A)? Given the matrices A and the reduced echelon form of A below, find: (b) A = 1234 5 1 0 1 2 -1 2246 4 1120 1 Consider vectors rref (A) = 1 0 1 0 0 1 1 0 0 0 0 1 00000 -5/3 8/3 1/3 V₁ = [1,-1, 5, 2], V2 = [−2, 3, 1, 0], V3 = [9, -10,34, 14], V4 = [0, 4, 2, -3], V5 = [-3, 31, 41, -8] in R4. Find a basis and the dimension of the subspace W = span{V1, V2, V3, V4, V5}. 10 705 0 1 -1 04 00 016 00 000 NOTE: RREF of the matrix whose columns are vectors V₁, V2, V3, V4, V5 is marks) Let A be an m x n matrix. Prove that every vector in the null space of A is orthogonal to every vector in the row space of A.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.2: Linear Independence, Basis, And Dimension

Problem 63EQ

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