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# General Vector Spaces problems in Linear Algebra

## Real Vector Space example question

Determine whether the given set equipped with the given operations is a vector space.

The set of all triplets of real numbers with the standard vector addition but with scalar multiplication defined by

k(x, y, z) = (k2x, k2y, k2z)

Solution to this Linear Algebra practice problem is given in the video below!

## Subspace example problem

Determine whether the following are subspaces of R3:

(a) All vectors of the form (a, 0, 0)

(b) All vectors of the form (a, 1, 0)

(c) All vectors of the form (a, b, c) where b = a + c

(d) All vectors of the form (a, b, c) where c = a b

(e) All vectors of the form (a, a, 0)

Solution to these Linear Algebra practice problems is given in the video below!

## Linear Combination example

Which of the following are linear combinations of u = (1, 3, 2) and v = (1, 0, 4)?

(a) (0, 3, 6)

(b) (3, 9, 2)

(c) (0, 0, 0)

(d) (1, 6, 16)

Solution to these Linear Algebra practice problems is given in the video below!

## Vector Span example question

Determine whether the given vector spans R3:

v1 = (1, 2, 3)

v2 = (2, 0, 0)

v3 = ( 2, 1, 0)

Solution to this Linear Algebra practice problem is given in the video below!

## Linear Independence example problem

Determine whether the following vectors in R4 are linearly dependent:

(1, 2, 2, 1)

(3, 6, 6, 3)

(4, 2, 4, 1)

Solution to this Linear Algebra practice problem is given in the video below!

## Basis of Vector Set example

Determine whether the following set of vectors form a basis in R3:

(1, 0, 0)

(2, 2, 0)

(3, 3, 3)

Solution to this Linear Algebra practice problem is given in the video below!

## Basis & Dimension of Solution Space example question

Find a basis for the solution space of the homogeneous linear system, and find the dimension of that space.

2x1 + x2 + 3x3 = 0

x1 + 5x3 = 0

x2 + x3 = 0

Solution to this Linear Algebra practice problem is given in the video below!

## Change of Basis example problem

Consider the bases

B = {u1, u2} and B’ = {u1‘, u2‘} for R2,

where

u1 = u2 = u1‘ = u2‘ = Find the transition matrix from B to B’.

Solution to this Linear Algebra practice problem is given in the video below!

## Subspace Basis example

Find a basis for the subspace of R4 spanned by the given vectors

(2, 4, 2, 3)

( 2, 2, 2, 4)

(1, 3, 1, 1)

Solution to this Linear Algebra practice problem is given in the video below!

## Rank of Matrix example question

Are there values of r and s for which the given matrix has rank 1 or rank 2? Solution to this Linear Algebra practice problem is given in the video below!