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Linear Algebra: A Modern Introduction

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter5: Orthogonality

Section5.4: Orthogonal Diagonalization Of Symmetric Matrices

Problem 26EQ

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Suppose that A is an nX m matrix of rank 3, the nullity of A is 4, and the column space of A is a subspace of R". Find the dimensions of A.
A has
rows and
columns.
Note: You can earn partial credit on this problem.
Prouiou Mu

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Suppose A is 10 by 10 and A2 = 0 (zero matrix). So A multiplies each column of A to give the zero vector. This means that the column space of A is contained in the __ . If A has rank r, those subspaces have dimension r � 10 � r. So the rank is '(' � 5.
Find a basis for the subspace of R3 spanned by S. S = {(5, 9, 9), (1, 2, 2), (1, 1, 1)}
Find a basis for the subspace of R3 spanned by S.S = {(4, 4, 8), (1, 1, 2), (1, 1, 1)}
1. Determine the exact distance between the planes 4 x + y – 2 z + 10 = 0 and 8 x + 2 y – 4 z – 12 = 0 . no matrix grade 12 knowloge only
We fit a ridge regression model to some data for λ=10 and λ=100 and we obtain the coefficients vector estimates b1 and b2 respectively. Then b1 is either more sparse than b2, or as sparse as b2. b2 is either more sparse than b1, or as sparse as b1. The L2 norm of b2 is less or equal to the L2 norm of b1. The L2 norm of b1 is less or equal to the L2 norm of b2.
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2. For each matrix A of Problem 1, write down orthonormal base for all four fundamental subspaces.(This can be read off from your answers to Problem 1.)
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Let B be a fixed 2×2 matrix, and letW = {A ∈ M2,2 | AB = BA}.Inotherwords,W isbethesetof2×2matricesAsuchthatAB=BA. ShowthatW isa subspace of M2,2.
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