MATLAB Ex 3 - Linear Space

MATLAB Exercise 3 – Linear Space
1. Which of the following are spanning sets for 31R ⨯? Justify your answers by at least two
different methods. (related function may be used: rank, rref, det )
1) 1010,1,0011⎧⎫⎛⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪ ⎪⎨⎬ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎩⎭
2) 10110,1,0,20113⎧⎫⎛⎫⎛⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪ ⎪ ⎪⎨⎬ ⎪ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭⎩⎭
3) 2321,2,2220⎧⎫⎛⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪ ⎪⎨⎬ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪--⎝⎭⎝⎭⎝⎭⎩⎭
4) 2241,1,2224⎧-⎫⎛⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪ ⎪-⎨⎬ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪--⎝⎭⎝⎭⎝⎭⎩⎭
2. Given
1213292,4,6,23265x x x y --⎛⎫⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪ ⎪====- ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭
1) Is 12Span(,)x x x ∈?
2) Is 12Span(,)y x x ∈?
(related function: rref, rank )
3. Which of the following are spanning sets for P 3? Justify your answer.
1) {
}
22
1,,1x x +
2) {
}
2
2,2,,21x x x -+
3) {
}
2
1,2,3x x x +-+
size(A,1) returns the number of rows of A
size(A,2) returns the number of columns of A
4. Which of the following collections of vectors are linearly independent?
1) 1011,1,0111⎧⎫⎛⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪ ⎪⎨⎬ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎩⎭
2) 221,220⎧⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪⎨⎬ ⎪ ⎪⎪⎪ ⎪ ⎪-⎝⎭⎝⎭⎩⎭
3) 2241,1,2224⎧-⎫⎛⎫⎛⎫⎛⎫⎪⎪ ⎪ ⎪ ⎪-⎨⎬ ⎪ ⎪ ⎪⎪⎪ ⎪ ⎪ ⎪--⎝⎭⎝⎭⎝⎭⎩⎭
5. For each of the sets of vectors in Exercise 4, describe geometrically the span of the given
vectors. (i.e. point out the maximal linearly independent subset for each of the sets of vectors.) 6. Given
1231302,4,103211x x x -⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪=== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭
1) Show that 123,,x x x are linearly dependent. 2) Show that 1x and 2x are linearly independent.
3) What is the dimension of Span (123,,x x x )? 4) Give a geometric description of Span (123,,x x x ). 7. The vectors
12345130212,4,10,7,3321132x x x x x --⎛⎫⎛⎫⎛⎫⎛⎫⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪===== ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭⎝⎭⎝⎭
Span R 3. Pare down the set 12345,,,,x x x x x to form a basis for R 3.
8. Let 123110,,237x x x ⎛⎫⎛⎫⎛⎫
=== ⎪ ⎪ ⎪⎝⎭⎝⎭⎝⎭
. Find the coordinates (坐标) of x with respect to 12,x x .
(i.e. A =(x 1,x 2), the coordinates of x with respect to 12,x x is c =A -1x ).
9. Let 1221,43x x ⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭
be a basis of R 2. Find the transition matrix from the standard basis {e 1,
e 2} to the basis {x 1, x 2} and determine the coordinates o
f 107x ⎛⎫
= ⎪⎝⎭
under the basis {x 1, x 2}.
10. Find the transition matrix corresponding to the change of basis from 12[,]x x to 12[,]y y ,
where 1221,43x x ⎛⎫⎛⎫== ⎪ ⎪⎝⎭⎝⎭, 1213,24y y -⎛⎫⎛⎫
== ⎪ ⎪⎝⎭⎝⎭
. (X -1 Y )
11. For each of the following matrices, find a basis for the row space （行空间）, a basis for the
column space （列空间）, and a basis for the null space (零空间={|0}x Ax =).
（If A is an m ×n matrix, the subspace of R 1×
n by the row vectors of A is called the row space
of A . The subspace of R m ×
1 by the column vectors of A is called the column space of A .）
1) 123236946A ⎛⎫ ⎪=- ⎪ ⎪⎝⎭
2) 12103
2562204A -⎛⎫
⎪
= ⎪ ⎪-⎝⎭ 3)12103
2562066A -⎛⎫
⎪
= ⎪ ⎪⎝⎭
12. *Generate 10×10, 100×100, 1000×1000 random matrices, and corresponding column
vectors. Compare the time used in solving linear system Ax = b by different functions or operators such as inv, \, lu and qr. (Refer to Lecture 3)
Matlab Help in the toolbar, then select Index and input qr ( and lu )to see different usage of these functions, understand what meaning of [L,U,P] = lu(A,0.0); [L,U,P] = lu(A,1.0); [Q,R] = qr(A,0) .
tic, operation, toc prints the number of seconds required for the operation.。