Week 3 Checkpoint

A short checkpoint on Week 3: vectors, the dot product, linear combinations, the L1/L2/L∞ norms, Euclidean distance, cosine similarity, and orthogonality. Aim for 80% or better before moving to Week 4. Each explanation names the chapter to revisit if you miss it.

  1. 1.

    Compute the dot product a⊤b\mathbf{a}^\top\mathbf{b} for a=[2,−1,3]\mathbf{a} = [2, -1, 3] and b=[4,5,1]\mathbf{b} = [4, 5, 1].

  2. 2.

    For u=[1,2]\mathbf{u} = [1, 2] and v=[3,0]\mathbf{v} = [3, 0], what is the linear combination 2u−v2\mathbf{u} - \mathbf{v}?

  3. 3.

    For x=[3,−4]\mathbf{x} = [3, -4], what are its L2 norm ∥x∥2\lVert\mathbf{x}\rVert_2 and its L1 norm ∥x∥1\lVert\mathbf{x}\rVert_1?

  4. 4.

    For x=[−2,6,−3]\mathbf{x} = [-2, 6, -3], what is the L∞ (max) norm ∥x∥∞\lVert\mathbf{x}\rVert_\infty?

  5. 5.

    Two nonzero vectors a\mathbf{a} and b\mathbf{b} satisfy a⊤b=0\mathbf{a}^\top\mathbf{b} = 0. What does this tell you?

  6. 6.

    The cosine similarity between a=[1,0]\mathbf{a} = [1, 0] and b=[1,1]\mathbf{b} = [1, 1] is a⊤b∥a∥ ∥b∥\dfrac{\mathbf{a}^\top\mathbf{b}}{\lVert\mathbf{a}\rVert\,\lVert\mathbf{b}\rVert}. What is its value?