Answered: 5. Consider the bases for R²: * - {^ -… | bartleby

Algebra & Trigonometry with Analytic Geometry

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.8: Determinants

Problem 8E

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5. Consider the bases for R²:
* - {^ - ( ) - - ( • ) } - {~ - () --()}
B₁ =
Οι =
√2 =
8
B2 =
=
=
(a) Find the transition matrix from B₁ to B2. Show all of your work.
(b) If u is a vector such that [u]B₁
=
(4)
find [B. Show all of your work.
(c) Find the transition matrix from B₂ to B₁

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Let B = {(1,3), (-2,-2)} and B' = {(-12,0), (-4,4)} be bases for R2, and let A= 4 3 0 2 be the matrix for T: R2 -> R2 relative to B
a) For invertible matrices A, BE Rª×", show that ((AB)")-1 = (A¯1)" (B')". b) Let A, B, C, DE RTX" be invertible matrices and suppose A" = C" ((A¯'D)")B. Solve for D in terms of A, B, C, their transposes, and their inverses. Your answer should not contain any nested expressions in parentheses.
U₁ = -D₁ --- --- -- = = V2 = (a) Find the transition matrix from the basis E = {v1, v2} to the basis F = {u₁, u₂} of R². (b) Use the transition matrix from (a) to write x = 5 from F. -4 [5] as a linear combination of the vectors
Let B = {(1, 3), (-2,-2)) and B' = {(-12, 0), (-4, 4)) be bases for R2, and let A = 02 be the matrix for T: R² R2 relative to B. (a) Find the transition matrix P from B' to B. P= (b) Use the matrices P and A to find [v]g and [7(V)]B, where [v] = [-14]. [v]8 = [T(v)]8 = (c) Find P¹ and A' (the matrix for 7 relative to B'). p-1 = A¹ = 1 1 ↓↑ (d) Find [T(v)] two ways.
then the standard matrix of T is A = 20 17 38 -17 20 T ([³]) = [²9] 38 ¹ ¹([^]) = [³] · and T
Let J, be then X n matrix whose entries are all equal to 1, and let D(^1, . , An) be the n X n diagonal matrix whose non-zero entries are 11, ... , An E R. Let x = (x1, ... , Xn) be a row vector in R' and let x' be its transpose, a column vector. 1. Show that xJ,x' > 0, for all x1, ... , Xn E R. 2. Assume that the scalars 11, ..., An are all strictly positive. Show that x D(11, ... , An)x' is strictly positive for all (x1, ..., Xn), unless x1 = •… Xn = 0. 3. Use questions 2 and 1 above to show that D(^1, .. , An) + 1Jn is an invertible matrix when 11, , ^n and 1 are all strictly positive.
please send handwritten solution for Q3
if A is (m*n) matrix, prove 'rank(A)=n-dim(N(A))

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