1 3 and let S be the set of vectors x E R³ such that Ax e Span { 2 (3) Let A = Find a basis for S 1 which contains the vector -3 1 Solution 1. Note that Ax e Span{} if and only if the first coordinate of Ax is 0, in other words if [1 1 3] x = [0], thus S = nul([1 1 3]). We find a basis for S by solving for the general solution in vector form; here the matrix is already in reduced echelon form, and x2, X3 are free variables; we solve for -x2 – 3x3 -31 xi to get x1 = -x2 3x3; thus the general solution is x = x2 = x2 1 + x3 To find X3 a basis containing note that it is contained in S (since [1 1 3|-3 0) so we can apply the 1 -3 1 procedure for finding a basis of the span of -3] 0 0 0 1 3 1 0 -1 1 1 -3 1 R2+=3R3 1 rearrange 1 3 1 1 1 1 0 0
1 3 and let S be the set of vectors x E R³ such that Ax e Span { 2 (3) Let A = Find a basis for S 1 which contains the vector -3 1 Solution 1. Note that Ax e Span{} if and only if the first coordinate of Ax is 0, in other words if [1 1 3] x = [0], thus S = nul([1 1 3]). We find a basis for S by solving for the general solution in vector form; here the matrix is already in reduced echelon form, and x2, X3 are free variables; we solve for -x2 – 3x3 -31 xi to get x1 = -x2 3x3; thus the general solution is x = x2 = x2 1 + x3 To find X3 a basis containing note that it is contained in S (since [1 1 3|-3 0) so we can apply the 1 -3 1 procedure for finding a basis of the span of -3] 0 0 0 1 3 1 0 -1 1 1 -3 1 R2+=3R3 1 rearrange 1 3 1 1 1 1 0 0
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter7: Eigenvalues And Eigenvectors
Section7.CM: Cumulative Review
Problem 18CM
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Equations and Inequations
Equations and inequalities describe the relationship between two mathematical expressions.
Linear Functions
A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.
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Can anyone explain why x2 and x3 is a free variable in the third line of the solution?
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