
Understanding Orthonormal Bases and the Gram-Schmidt Process

Interactive Video
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Mathematics
•
11th Grade - University
•
Hard

Aiden Montgomery
FREE Resource
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10 questions
Show all answers
1.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What is the primary advantage of having an orthonormal basis for a vector space?
It reduces the dimension of the vector space.
It makes the vectors linearly dependent.
It simplifies the calculation of vector projections.
It increases the number of vectors in the basis.
2.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
In a one-dimensional subspace, how can you ensure a vector is orthonormal?
By adding another vector to it.
By dividing the vector by its length.
By rotating the vector by 90 degrees.
By multiplying the vector by its length.
3.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What is the first step in constructing an orthonormal basis in a two-dimensional subspace?
Rotating the vectors to be perpendicular.
Adding a new vector to the subspace.
Normalizing the first vector.
Finding the cross product of the vectors.
4.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
How do you find the orthogonal component of a vector in a two-dimensional subspace?
By multiplying the vector by its projection.
By dividing the vector by its projection.
By adding the vector to its projection.
By subtracting the vector from its projection.
5.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What is the purpose of the Gram-Schmidt process?
To find the inverse of a matrix.
To create an orthonormal basis from a given basis.
To solve linear equations.
To reduce the dimension of a vector space.
6.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
In a three-dimensional subspace, what is the role of the vector y3?
It is the normalized version of V3.
It is the sum of U1 and U2.
It is the projection of V3 onto V2.
It is the orthogonal component of V3.
7.
MULTIPLE CHOICE QUESTION
30 sec • 1 pt
What is the next step after finding y3 in the Gram-Schmidt process?
Add y3 to U1 and U2.
Multiply y3 by V3.
Normalize y3 to get U3.
Subtract y3 from V3.
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