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!

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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!

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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!

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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!

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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!

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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!

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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!

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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!

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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!

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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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