Linear Algebra Answers

Let f:R²→R² be defined by f(x,y)=(-y,-x)

i) show that f is linear

ii)Determine a basis for the kernel of f and the nullity of f

iii) Determine the basis for the range of f and the rank of f

iv) Determine whether f is invertible or not

Construct an orthonormal basis for the subspace of R³ spanned by the vectors (1,-1,1)

and (2,0,4)

Determine the dimension and hence the basis for the vector space spanned by the vectors (-1,1,3),(2,3,4),(3,0,-5) and (-2,1,0)

Prove that if A and B are subspaces of Rⁿ, then AnB is also a subspace of Rⁿ

Define the linear function f:R³→R² by f(x,y,z)=(x-z,y-x,z-y).Find

i) the kernel of f

ii)the nulllity of f

iii)the rank of f and a basis for the range of f

Find the matrix representation A of the linear function: R²→R³ where

f(x,y)=(5x-y,2x-y,-x+2y) with respect to the standard bases for R² and R³

Consider the set B={(1,1,1),(0,2,2),(0,0,3)}.show that B

I) spans R³

ii)is linearly independent

iii)is a basis for R³

Show that the dim R⁵=5

Express the polynomial 1+3x+4x² as a linear combination of the polynomials

1+2x+3x², -1+x+x² and 2+x+x²