Understanding Subspaces and Projections in R3

The video tutorial explains the concept of a subspace V in R3, defined by vectors satisfying x1 + x2 + x3 = 0, which forms a plane. It covers finding the basis for this subspace and calculating the transformation matrix for projecting any vector in R3 onto V. The tutorial also explores the orthogonal complement of V and simplifies the projection matrix calculation using linear algebra techniques.

What is the defining equation for the subspace v in R3?

• x1 + x2 - x3 = 0

• x1 - x2 + x3 = 0

• x1 + x2 + x3 = 0

• x1 - x2 - x3 = 0

How can the subspace v be expressed using basis vectors?

• As the span of vectors (1, 0, 0) and (0, 1, 0)

• As the span of vectors (-1, 1, 0) and (-1, 0, 1)

• As the span of vectors (1, 1, 1) and (0, 0, 0)

• As the span of vectors (0, 0, 1) and (1, 1, 0)

What is the dimension of the matrix A used to find the projection of any vector x in R3 onto v?

• 4 by 2

• 3 by 3

• 2 by 3

• 3 by 2

What is the relationship between a vector in R3 and its projections onto a subspace and its orthogonal complement?

• The vector is unrelated to its projections onto the subspace and its orthogonal complement.

• The vector is the product of its projections onto the subspace and its orthogonal complement.

• The vector is the difference between its projections onto the subspace and its orthogonal complement.

• The vector is the sum of its projections onto the subspace and its orthogonal complement.

What is the orthogonal complement of the null space of a matrix equivalent to?

• The row space of the matrix

• The column space of the matrix

• The identity matrix

• The null space of the matrix

What is the result of multiplying a 3 by 1 matrix with a 1 by 3 matrix?

• A 1 by 3 matrix

• A 3 by 1 matrix

• A 3 by 3 matrix

• A 1 by 1 matrix

What is the inverse of a 1 by 1 matrix with an entry of 3?

• 1/3

• 3

• 0

• 1

What is the span of the vector (1, 1, 1) in R3?

• A point in R3

• A line in R3

• A plane in R3

• The entire R3 space

What is the transformation matrix for the projection onto v's orthogonal complement?

• 1/5 times a matrix of all 1's

• 1/4 times a matrix of all 1's

• 1/2 times a matrix of all 1's

• 1/3 times a matrix of all 1's

How is the transformation matrix for the projection onto the subspace v derived?

• By dividing the identity matrix by the transformation matrix for the orthogonal complement

• By adding the identity matrix to the transformation matrix for the orthogonal complement

• By subtracting the transformation matrix for the orthogonal complement from the identity matrix

• By multiplying the identity matrix with the transformation matrix for the orthogonal complement