John Rhodes, Math Teacher at Senioritis.io

John Rhodes
July 8, 2023
Linear Algebra

Proving the Existence of a Solution for Ax = b in Rn: The Importance of Spanning the Range of A

If A is an n x n matrix, then the equation Ax = b has at least one solution for each b in Rn. To show that...

John Rhodes
July 8, 2023
Linear Algebra

Understanding the Relationship Between the Span of Matrix A Columns and Linear Independence in Math

If the columns of A span R nm then the columns are linearly independent. To understand the relationship between the span of the columns of matrix A...

John Rhodes
July 8, 2023
Linear Algebra

Proving the Relationship Between the Trivial Solution and Row Equivalence to the Identity Matrix

If the equation Ax = 0 has only the trivial solution, then A is row equivalent to the n x n identity matrix. To prove that if...

John Rhodes
July 8, 2023
Linear Algebra

The Implications of Row Reducing a Square Matrix to the Identity Matrix and its Connection to Matrix Inverses

If square matrix A can be row reduced to the identity, then A must be invertible. If a square matrix A can be row reduced to the...

John Rhodes
July 8, 2023
Linear Algebra

Understanding Equality of Line Segments and Endpoints in Mathematics

In general, if AB = AC, then B = C. In general, if AB = AC, then we cannot conclude that B = C In general, if...

John Rhodes
July 8, 2023
Linear Algebra

Understanding Matrix Multiplication and Dimensions: Explaining why C, the product of matrices A and B, has the same number of columns as A and the same number of rows as B.

Suppose A, B, and C are matrices, and C = AB is defined. Then C has the same number of columns as A and the same number...

John Rhodes
July 8, 2023
Linear Algebra

Investigating the Properties of Matrix Multiplication: False Statement, Matrix AB = 0

If A and B are matrices, and the product AB is the zero matrix, then A or B must be a zero matrix. To determine if the...

John Rhodes
July 8, 2023
Linear Algebra

Proving C = D: All Corresponding Entries in C and D are Equal

If B, C, D, are square n x n invertible matrices and BC = BD, then C = D. To prove that C = D, we need...