Subspace Solution space Dimension of vector space subspace S

Subspace Solution space Dimension of vector space, subspace Span of a set of vectors Basis theorem (Always explain your answers.) What is the dimension of the subspace spanned by the following set of vectors: {[1 -2 4], [1 -3 5], [3 -7 13]} Express the subspace as a span of vectors from the set, using as few vectors as possible. What is the dimension of the subspace spanned by the following set of vectors: {[1 -2 -3], [1 -3 2], [2 0 1]} Express the subspace as a span of vectors from the set, using as few vectors as possible.

Solution

a =

1 1 3
-2 -3 -7
4 5 13

determinant of a = 0

dimension is less than 3.

v1 = [1 -2 4]; v2 = [1 -3 5]

are independent to each other

the dimension is 2 .

subspace can be written as span of v1 and v2

2)

b =

1 1 2
-2 -3 0
-3 2 1

determinant is -27.0000 ,which is non-zero

hence dimension = 3

subspace will be spanned by all column vectors