MA 305 EXAM 3 REVIEW QUESTIONS
What does the norm represent? - Answers -The length of a vector
To calculate the distance between 2 vectors x and y... - Answers -Subtract each
corresponding component from x and y, square each result, and take the square root of
the entire sum of the results. (Distance Formula / Subtraction of components and
compute Norm)
|| x - y || = || y - x ||
State the formula for the Inner Product of 2 Vectors - Answers -< x, y > OR x^Ty OR
y^Tx
State the formula for the angle between 2 Vectors - Answers -The inverse cosine of the
inner product over the norms of the vectors multiplied
Θ = cos^(-1)((< x, y >) / (|| x || || y ||))
State the Cauchy-Schwarz Inequality - Answers -|< x, y >| ≤ || x || || y ||
Two vectors are orthogonal if... - Answers -Their inner product is 0
< x, y > = 0
The zero vector is orthogonal to every other vector, true or false? - Answers -True
State the definition of a Norm - Answers -The square root of the inner product of the
vector to itself.
|| x || = sqrt(< x, x >)
The Norm represents the length of a vector, true or false? - Answers -True
There are not other ways to define norms other than || x || = sqrt(< x, x >), true or false?
- Answers -False
The distance between 2 vectors can be computed as || x - y || OR || y - x || - Answers -
True
State the Properties of Norms (Hint: There are 3) - Answers -1. Positivity
2. Homogeneity
3. Triangle Inequality
State the Positivity Property of Norms - Answers -|| x || ≥ 0
What does the norm represent? - Answers -The length of a vector
To calculate the distance between 2 vectors x and y... - Answers -Subtract each
corresponding component from x and y, square each result, and take the square root of
the entire sum of the results. (Distance Formula / Subtraction of components and
compute Norm)
|| x - y || = || y - x ||
State the formula for the Inner Product of 2 Vectors - Answers -< x, y > OR x^Ty OR
y^Tx
State the formula for the angle between 2 Vectors - Answers -The inverse cosine of the
inner product over the norms of the vectors multiplied
Θ = cos^(-1)((< x, y >) / (|| x || || y ||))
State the Cauchy-Schwarz Inequality - Answers -|< x, y >| ≤ || x || || y ||
Two vectors are orthogonal if... - Answers -Their inner product is 0
< x, y > = 0
The zero vector is orthogonal to every other vector, true or false? - Answers -True
State the definition of a Norm - Answers -The square root of the inner product of the
vector to itself.
|| x || = sqrt(< x, x >)
The Norm represents the length of a vector, true or false? - Answers -True
There are not other ways to define norms other than || x || = sqrt(< x, x >), true or false?
- Answers -False
The distance between 2 vectors can be computed as || x - y || OR || y - x || - Answers -
True
State the Properties of Norms (Hint: There are 3) - Answers -1. Positivity
2. Homogeneity
3. Triangle Inequality
State the Positivity Property of Norms - Answers -|| x || ≥ 0
