Linear Algebra: Subspaces, Kernels, and Matrix Maps

Question 6: Vector Spaces and Subspaces

Finding the Zero Vector

1. Take the general scalar multiplication formula r ◦ (a, b, c, d).
2. Plug in r = 0.
3. Read off the resulting matrix; that is the zero vector. (Note: 0 ◦ anything = the zero vector in any vector space.)

Proving a Subset U is a Subspace

1. Identify which entries of U are locked (fixed numbers) and which are free (variables).
2. Check if the zero vector (from part a) fits the locked entries of U.
3. Take two general elements of U (using the same locked numbers but different free letters) and combine them with addition. Check that the locked entries stay locked.
4. Take one general element of U and scale it with ◦. Check that the locked entries stay locked.
5. If all three subspace conditions hold, then U is a subspace.

Determining if Two Vectors are Dependent

1. Write the general formula for r ◦ v₁ using the given scalar multiplication.
2. Set r ◦ v₁ equal to v₂, matching entry by entry. This gives four equations.
3. Solve each equation for r.
4. If all four equations give the same r, the vectors are dependent (write v₂ = r ◦ v₁). If they conflict, they are independent.

Question 8: Linear Transformations and Matrices

Finding the Kernel (ker(h))

1. Set h(input) equal to the zero of the codomain (every coefficient or component equals 0).
2. This gives one equation per coefficient or component.
3. Solve the system for the input variables.
4. The ker(h) is the set of those solutions. If only the trivial solution exists, then ker = {0} and h is injective.

Determining the Range (Range(g))

1. Apply g to every vector of the domain's standard basis (e.g., 1, x, x² or e₁, e₂).
2. The Range(g) is the span of those images.
3. Check the independence of the images:
   • If the matrix is square (same count as codomain dimension), take the determinant. A nonzero result means they are independent, and the Range is the whole codomain (g is onto).
   • If the determinant is 0 (or the matrix is not square), the Range is a smaller subspace.

Calculating the Composition (g ∘ h)

1. From the formula for h, write the output coefficients or components in terms of the original input variables (a, b, ...). Call these a₀, a₁, a₂, etc.
2. Substitute these expressions into the formula for g wherever it uses a₀, a₁, a₂.
3. Simplify each component algebraically by collecting like terms.
4. Write the final result as (g ∘ h)(input) = (component 1, component 2, ...).

Matrix Representations

1. Optional shortcut: Derive a formula converting any output into C-coordinates by matching coefficients once on a general polynomial or vector.
2. Take each vector in basis B, one at a time, and plug it into the formula for h to get a raw output.
3. Convert that raw output into C-coordinates (using the shortcut or by matching coefficients directly). For example: 3 + x + 2x² = α(1+x+x²) + β(x+x²) + γ(x²).
4. That coordinate vector forms one column of the matrix.
5. Repeat for every vector in B and stack the columns in the order of basis B.
6. Check the shape: rows = dim(codomain) and columns = dim(domain).

Matrix of a Composition

1. Multiply the two matrices from the previous part, ensuring the inner map (h) is on the right: Rep_{B,D}(g ∘ h) = Rep_{C,D}(g) · Rep_{B,C}(h).
2. Perform the matrix multiplication entry by entry (row · column).