QUESTION: Let U be a finite dimensional vector space over F with basis By : U1,….., Up. Let SE L(W,U). Just using the definitions and results above, prove that: col,(M(ST)) = col;(M(S)M(T)),j=1,..., m. Hence M(ST) = M(S)M(T). Furthermore, for any v EV: [STv]BU = M(STv) = M(ST)M(v) = M(S)M(T)[v]Bv-
QUESTION: Let U be a finite dimensional vector space over F with basis By : U1,….., Up. Let SE L(W,U). Just using the definitions and results above, prove that: col,(M(ST)) = col;(M(S)M(T)),j=1,..., m. Hence M(ST) = M(S)M(T). Furthermore, for any v EV: [STv]BU = M(STv) = M(ST)M(v) = M(S)M(T)[v]Bv-
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![First recall the following notation and results discussed in class:
col,(A) denotes the jth column of an n x m matrix A with entries from F, that is, col,(A) E F, the set of
column vectors with n rows.
For any p x n matrix B and any n x m matrix A, the definition of matrix multiplication gives that
col, (BA) — В соl,(А), j — 1, 2, ..., т.
In the special case that A is a column vector (ie m = 1) with entries 21,..., En , then the above gives that
BA =x,col1 (B) + x2col2(B)+.+¤ncol,(B).
Given a basis By : V1,..., Vn for a vector space Vover F, for any v E V , the "matrix of v" given in Definition 3.62
of the textbook is also called the "coordinate vector of v with respect to the basis By", that is
[v)Bv = M(v) E F,1.
Given two finite dimensional vector spaces V and W over F and any linear map T E L(V,W), let By : v1,..., Vn
be a basis for V and Bw : w1,..., wm be a basis for W. Then from Definition 3.32 of the textbook for the matrix of
T we know that:
col,(M(T)) = [Tv;]Bw= M(Tv;).
In class, using the above properties/results, we proved Result 3.65 from the textbook, for any v E V:
[Tu]Bw = M(Tv) = M(T)M(v) = M(T)[v]Bv-
QUESTION: Let U be a finite dimensional vector space over F with basis BU : u1,..., Up. Let SE L(W,U). Just
using the definitions and results above, prove that:
col,(M(ST)) = col;(M(S)M(T)), j= 1,..., m.
Hence M(ST) = M(S)M(T). Furthermore, for any v E V:
[STv]BU = M(STv) = M(ST)M(v) = M(S)M(T)[v]Bv-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb39617aa-1a85-4ff5-98c6-3f7fe0e80deb%2Feaded6c1-ed6b-4672-8c7d-28a5c721c153%2F4q6qhee_processed.png&w=3840&q=75)
Transcribed Image Text:First recall the following notation and results discussed in class:
col,(A) denotes the jth column of an n x m matrix A with entries from F, that is, col,(A) E F, the set of
column vectors with n rows.
For any p x n matrix B and any n x m matrix A, the definition of matrix multiplication gives that
col, (BA) — В соl,(А), j — 1, 2, ..., т.
In the special case that A is a column vector (ie m = 1) with entries 21,..., En , then the above gives that
BA =x,col1 (B) + x2col2(B)+.+¤ncol,(B).
Given a basis By : V1,..., Vn for a vector space Vover F, for any v E V , the "matrix of v" given in Definition 3.62
of the textbook is also called the "coordinate vector of v with respect to the basis By", that is
[v)Bv = M(v) E F,1.
Given two finite dimensional vector spaces V and W over F and any linear map T E L(V,W), let By : v1,..., Vn
be a basis for V and Bw : w1,..., wm be a basis for W. Then from Definition 3.32 of the textbook for the matrix of
T we know that:
col,(M(T)) = [Tv;]Bw= M(Tv;).
In class, using the above properties/results, we proved Result 3.65 from the textbook, for any v E V:
[Tu]Bw = M(Tv) = M(T)M(v) = M(T)[v]Bv-
QUESTION: Let U be a finite dimensional vector space over F with basis BU : u1,..., Up. Let SE L(W,U). Just
using the definitions and results above, prove that:
col,(M(ST)) = col;(M(S)M(T)), j= 1,..., m.
Hence M(ST) = M(S)M(T). Furthermore, for any v E V:
[STv]BU = M(STv) = M(ST)M(v) = M(S)M(T)[v]Bv-
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