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Linear Algebra - Khan Academy

Linear Algebra - Khan Academy

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Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.

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What a matrix is. How to add and subtract them.

What the inverse of a matrix is. Examples of inverting a 2x2 matrix. Practice this lesson yourself on right now: ...

Inverting a 3x3 matrix Practice this lesson yourself on right now: ...

Using Gauss-Jordan elimination to invert a 3x3 matrix. Practice this lesson yourself on right now: ...

Using the inverse of a matrix to solve a system of equations.

Using matrices to figure out if some combination of 2 vectors can create a 3rd vector

When and why you can't invert a matrix. Practice this lesson yourself on right now: ...

Visual intuition of a 3-variable linear equation.

Solving 3 equations with 3 unknowns (old video from July 2008)

Visually understanding basic vector operations Practice this lesson yourself on right now: ...

More examples determining linear dependence or independence.

Determining whether 3 vectors are linearly independent and/or span R3

Understanding the definition of a basis of a subspace

Proving the "associative", "distributive" and "commutative" properties for vector dot products.

Determining the equation for a plane in R3 using a point on the plane and a normal vector

Proof: Relationship between the cross product and sin of angle between vectors

Solving a system of linear equations by putting an augmented matrix into reduced row echelon form

Another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form

And another example of solving a system of linear equations by putting an augmented matrix into reduced row echelon form

Defining and understanding what it means to take the product of a matrix and a vector

Understanding how the null space of a matrix relates to the linear independence of its column vectors

Figuring out the null space and a basis of a column space for a matrix

Showing that linear independence of pivot columns implies linear independence of the corresponding columns in the original equation

Showing that just the columns of A associated with the pivot columns of rref(A) do indeed span C(A).

Showing how ANY linear transformation can be represented as a matrix vector product

Exploring what happens to a subset of the domain under a transformation

Showing that the image of a subspace under a transformation is also a subspace. Definition of the image of a Transformation.

Example involving the preimage of a set under a transformation. Definition of kernel of a transformation.

Sums and Scalar Multiples of Linear Transformations. Definitions of matrix addition and scalar multiplication.

Creating scaling and reflection transformation matrices (which are diagonal)

What unit vectors are and how to construct them

Proof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f.

Relating invertibility to being onto (surjective) and one-to-one (injective)

Showing that the rank of the of an mxn transformation matrix has to be an for the transformation to be one-to-one (injective)

Showing that a transformation is invertible if and only if rref(A) is equal to the identity matrix

Determining a method for constructing inverse transformation matrices

Figuring out the formula for a 2x2 matrix. Defining the determinant.

Determinants: Finding the determinant of a 3x3 matrix

Defining the determinant for nxn matrices. An example of a 4x4 determinant.

A alternative "short cut" for calculating 3x3 determinants (Rule of Sarrus)

Correction of last video showing that the determinant when one row is multiplied by a scalar is equal to the scalar times the determinant

The determinant when one matrix has a row that is the sum of the rows of other matrices (and every other term is identical in the 3 matrices)

What happens to the determinant when we perform a row operation

Calculating a 4x4 determinant by putting in in upper triangular form first.

Realizing that the determinant of a 2x2 matrix is equal to the area of the parallelogram defined by the column vectors of the matrix

Viewing the determinant of the transformation matrix as a scaling factor of regions

Proof by induction that transposing a matrix does not change its determinant

Transpose of a column vector. Matrix-matrix products using vectors

Relationship between left nullspace, rowspace, column space and nullspace.

Showing that if V is a subspace of Rn, then dim(V) + dim(V's orthogonal complement) = n


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