# Linear Algebra Examples

,

Step 1

Step 1.1

The dot product of two vectors is the sum of the products of the their components.

Step 1.2

Simplify.

Step 1.2.1

Simplify each term.

Step 1.2.1.1

Multiply by .

Step 1.2.1.2

Multiply by .

Step 1.2.1.3

Multiply by .

Step 1.2.2

Add and .

Step 1.2.3

Add and .

Step 2

Step 2.1

The norm is the square root of the sum of squares of each element in the vector.

Step 2.2

Simplify.

Step 2.2.1

One to any power is one.

Step 2.2.2

Raise to the power of .

Step 2.2.3

One to any power is one.

Step 2.2.4

Add and .

Step 2.2.5

Add and .

Step 3

Find the projection of onto using the projection formula.

Step 4

Substitute for .

Step 5

Substitute for .

Step 6

Substitute for .

Step 7

Step 7.1

Rewrite as .

Step 7.1.1

Use to rewrite as .

Step 7.1.2

Apply the power rule and multiply exponents, .

Step 7.1.3

Combine and .

Step 7.1.4

Cancel the common factor of .

Step 7.1.4.1

Cancel the common factor.

Step 7.1.4.2

Rewrite the expression.

Step 7.1.5

Evaluate the exponent.

Step 7.2

Cancel the common factor of and .

Step 7.2.1

Factor out of .

Step 7.2.2

Cancel the common factors.

Step 7.2.2.1

Factor out of .

Step 7.2.2.2

Cancel the common factor.

Step 7.2.2.3

Rewrite the expression.

Step 7.3

Multiply by each element of the matrix.

Step 7.4

Simplify each element in the matrix.

Step 7.4.1

Multiply by .

Step 7.4.2

Multiply .

Step 7.4.2.1

Combine and .

Step 7.4.2.2

Multiply by .

Step 7.4.3

Multiply by .