Ch.1.4 - The Matrix Equation

Ax = b
What is the matrix equation?
To view linear combination of vectors as the product of a matrix and a vector.
What is the fundamental idea of linear algebra?
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What is DEFINITION 1.4?
What is DEFINITION 1.4?
Only if the number of columns equals the number of entries in x.
When the matrix equation defined?
A matrix equation, as a vector equation, or as a system of linear equations.
What are the three equivalent ways a system of linear equations can be viewed as?
What is THEOREM 3?
What is THEOREM 3?
The equation ?? = ? has a solution i.f.f. ? is a linear combination of the columns ?.
How do you know if a solution exists (existence of solutions) ?
“Is ? in Span {a₁, . . . an} ?”

Similar was of showing that it is or isn’t consistent.

What is the question “Is ?? = ? consistent?” equivalent to from 1.3.?
To determine whether the equation “?? = ?” is consistent for all possible ?.
What is a harder existence problem?
What is THEOREM 4?
Only for a coefficient matrix, if the augmented matrix [ ? ?] has a pivot position in every row, then the equation ?? = ? may or may not be consistent.
What is THEOREM 4?
Use the dot product that corresponds with entry one in A and vector one in X.

e.g. 2•X₁ + 3•X₂

When calculating entries for ?? by hand, what is a tip to keeping in mind?
What is the Row-Vector Rule for computing ???
If the product ?? is defined, then the ? th entry in ?? is the sum of the products of corresponding entries from row ? of ? and from the vector x.
What is the Row-Vector Rule for computing ???
What is an identity matrix?
When there are only 1 = b entries within the matrix. Hence, they are the identity for the values of x…
What is an identity matrix?
What is THEOREM 5?
What is THEOREM 5?
The sequence of calculations should involve data stored in the contiguous memory locations.
To optimize a computer algorithm to compute ???
Fortran, which is a language that stores a matrix as a set of columns.

-Such algorithms compute ?? as a linear combination the columns of ?.

What are the most widely used professional algorithms written in?
The C language stores matrices by rows, and ?? should be computed by the alternative rule that uses the ROWS of ?.
What if a program were written in C?
There will be ? number of leading entries which implies ? number of pivot points. If n does not equal n then it is now within the span. If however n does equal n, then then columns do span ℝⁿ.
How do we know if the columns in the matrices span ℝⁿ?
It must have at least one solution.
How does this new language fit into the question of whether a linear systems of equations is consistent?
I.f.f. ? is a linear combination of the columns of ?.
When does the equation Is ?? = ? have a solution?
In is a n x n matrix with 1’s on the diagonal , and 0’s everywhere else.
What is the identity matrix?

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