Linear Algebra

Use this for “Reduce [A | I] to find A⁻¹ for A = [[2, 1, 0], [1, 1, 0], [0, 0, 3]].”

Use this for “Factor A = [[2, 1, 1], [4, 3, 3], [8, 7, 9]] as A = LU.”

Use this for “Is [[1, 3, 4], [0, 1, 2]] in echelon form, reduced echelon form, or neither?”

Use this for “Are ⟨1, 0, 1⟩, ⟨2, 1, 0⟩ and ⟨0, 1, 1⟩ linearly independent?”

Use this for “Find the coordinates of x = ⟨5, 1⟩ in the basis b₁ = ⟨1, 1⟩, b₂ = ⟨1, −1⟩.”

Use this for “Is the set of points with y = 3x − 2 a subspace of ℝ²?”

Use this for “Find the volume of the box with edges ⟨2, 0, 0⟩, ⟨1, 3, 0⟩ and ⟨0, 1, 4⟩.”

Use this for “A and B are 3 × 3 with det A = 4 and det B = −2. Find det(2A), det(AB), det A⁻¹ and det Aᵀ.”

Use this for “λ = 3 is an eigenvalue of [[2, 0, 0], [1, 3, 0], [0, 1, 1]]. Find an eigenvector for it.”

Use this for “A = [[4, 1], [2, 3]] and x = (1, 0). Find A³x by writing x in the eigenvectors of A.”

Use this for “Each year 10% of city dwellers move to the suburbs and 30% of suburb dwellers move to the city. What share lives in each in the long run?”

Use this for “Find the eigenvalues of [[1, −1], [1, 1]] and the angle and scale factor of the map.”

Use this for “Find the projection of u = ⟨3, 1⟩ onto v = ⟨1, 1⟩ and the part of u orthogonal to v.”

Use this for “Apply Gram–Schmidt to v₁ = ⟨1, 1, 0⟩ and v₂ = ⟨1, 0, 1⟩ to get an orthogonal basis of their plane.”

Use this for “T(x) = Ax with A = [[1, 2, −1], [2, 4, 1], [3, 6, 0]]. Find the kernel and the range of T. Is T one-to-one? Onto?”

Use this for “Rotate (3, 1) by 90° and then reflect it in the x-axis. Find the image and the one matrix that does both.”

Use this for “Find the singular values of A = [[3, 0], [4, 5]].”