Untitled Flashcards Set

Front: What is the definition of the inner product (dot product) of vectors v and w in ℝⁿ?
Back: v⋅w=v1w1+v2w2+⋯+vnwnv⋅w=v1​w1​+v2​w2​+⋯+vn​wn​. Equivalent to vTwvTw for column vectors.

Front: Why is the inner product commutative (v⋅w=w⋅vv⋅w=w⋅v)?
Back: Transposing a scalar doesn’t change it: vTw=(vTw)T=wTvvTw=(vTw)T=wTv.

Front: When is v⋅v=0v⋅v=0?
Back: Only when v is the zero vector (all components are 0).

Front: How is the norm (length) of v defined?
Back: ∥v∥=v⋅v=v12+v22+⋯+vn2∥v∥=v⋅v​=v12​+v22​+⋯+vn2​​.

Front: Define the distance between vectors v and w.
Back: dist(v,w)=∥v−w∥dist(v,w)=∥v−w∥.

Front: What does it mean for vectors v and w to be orthogonal?
Back: Their inner product is zero: v⋅w=0v⋅w=0.

Front: What is an orthogonal set?
Back: A set of vectors where every distinct pair has an inner product of 0.

Front: How do you normalize a non-zero vector v?
Back: Divide by its norm: u=v∥v∥u=∥v∥v​.

Front: What is an orthonormal set?
Back: A set of unit vectors that are pairwise orthogonal.

Front: In ℝ²/ℝ³, how are orthogonality and perpendicularity related?
Back: Orthogonal vectors are geometrically perpendicular (Theorem 23, proven via the Pythagorean theorem).

Front: Are [1, 1] and [1, -1] orthogonal?
Back: Yes: 1⋅1+1⋅(−1)=01⋅1+1⋅(−1)=0.

Front: Normalize v = [1, 2].
Back: u=[15,25]u=[5​1​,5​2​].

Front: Convert [1, 1] and [1, -1] to an orthonormal set.
Back: Normalize: [12,12][2​1​,2​1​] and [12,−12][2​1​,2​−1​].

Front: What’s the error in calculating dist(v=[2,-1,1], w=[0,1,0])?
Back: Original answer was 66​. Correct: (2−0)2+(−1−1)2+(1−0)2=3(2−0)2+(−1−1)2+(1−0)2​=3.

Front: List the four inner product properties (Theorem 22).
Back:

1. Commutativity

2. Distributivity: (u+v)⋅w=u⋅w+v⋅w(u+v)⋅w=u⋅w+v⋅w

3. Scalar multiplication: (cu)⋅v=c(u⋅v)(cu)⋅v=c(u⋅v)

4. v⋅v≥0v⋅v≥0, and equals 0 only if v = 0.

Front: What’s the first step to check orthogonality?
Back: Compute the dot product. If it’s 0, they’re orthogonal.

Front: How to find w orthogonal to u=[1,1,0] and v=[1,-1,0]?
Back: Solve w1+w2=0w1​+w2​=0 and w1−w2=0w1​−w2​=0. Solution: w = [0, 0, k] (any non-zero k).

Front: State the Pythagorean theorem for orthogonal v and w.
Back: ∥v+w∥2=∥v∥2+∥w∥2∥v+w∥2=∥v∥2+∥w∥2.

Front: Why is v⋅v≥0v⋅v≥0?
Back: It’s a sum of squares, which are always non-negative.

These cards cover definitions, properties, examples, and problem-solving strategies for inner products and orthogonality.