MATRIX MULTIPLICATION RESULT

{{ outputs.mul[0] }}	{{ outputs.mul[1] }}	{{ outputs.mul[2] }}
{{ outputs.mul[3] }}	{{ outputs.mul[4] }}	{{ outputs.mul[5] }}
{{ outputs.mul[6] }}	{{ outputs.mul[7] }}	{{ outputs.mul[8] }}

To multiply a matrix by another matrix we need to do the dot product of rows and columns.

The "Dot Product" is where we multiply matching members, then sum up:
(1, 2, 3) • (7, 9, 11) = 1×7 + 2×9 + 3×11
 = 58
We match the 1st members (1 and 7), multiply them, likewise for the 2nd members (2 and 9) and the 3rd members (3 and 11), and finally sum them up.

We can do the same thing for the rest of the rows and columns to get:

MATRIX INVERSE RESULT

{% if outputs.inv=="Inverse Does not exist"%}

INVERSE DOES NOT EXIST FOR THE GIVEN MATRIX

{% else %}

{{ outputs.inv[0] }}	{{ outputs.inv[1] }}	{{ outputs.inv[2] }}
{{ outputs.inv[3] }}	{{ outputs.inv[4] }}	{{ outputs.inv[5] }}
{{ outputs.inv[6] }}	{{ outputs.inv[7] }}	{{ outputs.inv[8] }}

{% endif %}

Just like how a number has a reciprocal, a matrix has its inverse.

When we multiply a matrix by its inverse we get the Identity Matrix (which is like "1" for matrices): A × A^-1 = I

where I is the Identity Matrix.

MATRIX NULL SPACE RESULT

{% if outputs.nul=="Empty set"%}

NULL SPACE IS AN EMPTY SET

{% endif %} {% for i in range(outputs.nulcount/3) %}

Null Space Basis {{ i+1 }}: {{ outputs.nul[3*i] }} {{ outputs.nul[3*i+1] }} {{ outputs.nul[3*i+2] }}

{% endfor %}

The null space of a matrix A is the set of vectors that satisfy the homogeneous equation Ax = 0. The size of the null space of the matrix provides us with the number of linear relations among attributes.

IDEA:

Ax = 0 implies every row of A when multiplied by xgoesto zero.
Variable values in each sample(represented by a row) behave the same.
This helps in identifying the linear relationships in the attributes.
Every null space vector corresponds to one linear relationship.

Nullity can be defined as the number of vectors present in the null space of a given matrix. In other words, the dimension of the null space of the matrix A is called the nullity of A.

MATRIX COLUMN SPACE RESULT

{% if outputs.col=="Empty set"%}

COLUMN SPACE IS AN EMPTY SET

{% endif %} {% for i in range(outputs.colcount/3) %}

Column Space Basis{{i+1}}: {{ outputs.col[3*i] }} {{ outputs.col[3*i+1] }} {{ outputs.col[3*i+2] }}

{% endfor %}

What is Column Space?

A column space (or range) of matrix X is the space that is spanned by X’s columns.

If any column is a linear combination of other columns, it will not be part of the basis for column space.

MATRIX ROW SPACE RESULT

{% if outputs.row=="Empty set"%}

ROW SPACE IS AN EMPTY SET

{% endif %} {% for i in range(outputs.rowcount/3) %}

Row Space Basis {{ i+1 }}: {{ outputs.row[3*i] }} {{ outputs.row[3*i+1] }} {{ outputs.row[3*i+2] }}

{% endfor %}

The vector space generated by the rows of a matrix viewed as vectors. The row space of a n×m matrix A with real entries is a subspace generated by n elements of R^m, hence its dimension is at most equal to min(m,n). It is equal to the dimension of the column space of A, and is called the rank of A.

Similar to columnspace basis, the row space basis contains vectors who are linearly independant of each other.

QR DECOMPOSITION RESULT

Q Matrix:

{{ outputs.Q[0] }}	{{ outputs.Q[1] }}	{{ outputs.Q[2] }}
{{ outputs.Q[3] }}	{{ outputs.Q[4] }}	{{ outputs.Q[5] }}
{{ outputs.Q[6] }}	{{ outputs.Q[7] }}	{{ outputs.Q[8] }}

R Matrix:

{{ outputs.R[0] }}	{{ outputs.R[1] }}	{{ outputs.R[2] }}
{{ outputs.R[3] }}	{{ outputs.R[4] }}	{{ outputs.R[5] }}
{{ outputs.R[6] }}	{{ outputs.R[7] }}	{{ outputs.R[8] }}

The QR decomposition (or QR factorization) allows us to express a matrix having linearly independent columns as the product of 1) a matrix Q having orthonormal columns and 2) an upper triangular matrix R.

To understand QR decomposition, it is needed to know what the Gram-Schmidt orthonormalization process does.

It basically transforms a set of linearly independent vectors to a set of orthonormal vectors that span the same space spanned by the original set. In the QR decomposition, the matrix Q is the result obtained from Gram-Schmidt.

LINEAR TRANSFORMATION RESULT

{% if outputs.eigenvectors=="Empty set"%}

NO EIGENVECTORS

{% endif %} {% for i in range(outputs.eigsize) %}

Eigenvector {{ i+1 }}: {{ outputs.eigenvectors[3*i] }}, {{ outputs.eigenvectors[3*i+1] }}, {{ outputs.eigenvectors[3*i+2] }} Eigenvalue : {{ outputs.eigenvalues[i] }}

{% endfor %}

Eigenvectors are those vectors whose span does not change even after a linear transformation. But they get scaled upon transformation, by a parameter called an eigenvalue. Each eigenvector has a corresponding eigenvalue for a particular linear transformation.

The original vector is:

{{ outputs.original_vector[0] }} {{ outputs.original_vector[1] }} {{ outputs.original_vector[2] }}

The transformed vector is:

{{ outputs.transformed[0] }} {{ outputs.transformed[1] }} {{ outputs.transformed[2] }}

{{ outputs.plot }}

Linear Algebra Genie