Linear Algebra Plus Worksheet, Exercises of Mathematics

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Double PLUS Session Worksheet

MATH 1554: Linear Algebra

Fall 2024

Adapted from work of Nora Butler, Ahmed Mohammed, and Sri Palaniappan

Double 1, Fall 2023 T/F Questions 1

1 T/F Questions

Question 1 For each statement, mark the statement true if it’s always true, or false otherwise.

true false

Given the linear T : Rn^ → Rm^ where m = n and T (x) = b has a solution for every b ∈ Rm, T is one-to-one.

If A ∈ Rm×n^ and {b 1 , b 2 , b 3 ,... , bp} is the set of vectors from Rm^ such that for any vector bi in the set, Ax = bi has a solution, then Ax = ˜b, where ˜b ∈ Span{b 1 , b 2 , b 3 ,... , bp}, also has a solution.

If a matrix has a mixture of pivot and non-pivot columns, then the non-pivot columns can be expressed as a linear combination of the pivot columns.

For any two matrices A, B ∈ Rn×n, where A ̸= B, AB ̸= BA.

If the columns of a 16 × 42 matrix A span all of R^16 , then there exists 26 free variables in the equation Ax = b.

It is possible that a set of vectors {v 1 , v 2 , v 3 } in R^3 form a linearly indpenendent set but not span all of R^3.

There exists a vector b ∈ R^2 such that the solution set of the equation  1 3 0 0 0 1

x 1 x 2 x 3

 (^) = b is the x 3 axis.

2 Possible/Impossible Questions

Question 2 Mark each statement as possible if it could ever be true, or impossible otherwise.

possible impossible

A linear transformation T : R^2 → R^3 that is one-to-one and its standard matrix has a pivot in every column. A linear transformation T : R^3 → R^2 that is onto but not one-to-one.

A linear transformation T : R^3 → R^2 that is not onto and not one-to-one.

A linearly dependent set made up of two vectors, but one of them is not a scalar multiple of the other.

A matrix A ∈ Rm×n^ such that the equation Ax = b has at least one solution for an infinite number of vectors b ∈ Rm^ and also the equation Ax = d has no solutions for an infinte number of vectors d ∈ Rm.

A matrix A ∈ Rm×n^ with m < n and A has a pivot in every column.

A matrix A where the equation Ax = b has multiple solutions but only the trivial solution is a solution to the equation Ax = 0.

A set of linearly independent vectors from Rn^ {v 1 , v 2 , v 3 } where v 3 ∈ Span{v 1 , v 2 }.