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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!






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