Linear Algebra: Basis and Dimension

  • #1

Homework Statement


In each case, check that (v1,....vk) is a basis for Rn and give the coordinates of the given vector b belongs to Rn with respect to that basis.


Homework Equations


a) v1=(2,3) v2=(3,5) b=(3,4)
b) v1=(1,0,1) v2=(1,1,2) v3=(1,1,1) b=(3,0,1)

The Attempt at a Solution


I put it into a matrix and row reduced the matrix and got:
a) column 1 = (1,0) column 2 = (0,1) equals column 3 (3, -1)
Then I said the coordinates were thus (3,-1)
b) I did the same thing and got:
column 1 = (1,0,0) column 2 = (0,1,0) column 3=(0,0,1) equals column 4 =(-1, -2, 2)
The coordinates are (-1,-2,2)


Question 2

Homework Statement


Find a basis for each of the given subspaces and determine its dimension.


Homework Equations


a) V={x belonging to R4: x1+x2+x3+x4=0, x2+x4=0} c R4
b) V=(Span(1,2,3))orthogonal c R3

The Attempt at a Solution


I put it into a matrix and row reduced the matrix and got:
a) column 1 = (1,0, 0, 0) column 2 = (0,1,0,0) So I said the dimension was 2 and the basis was {(1,0)(1,1)}
b) For part two I don't know how to start this because I do not know what the matrix will look like.
 

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