Understanding Vector Span in R3

The video tutorial explains the concept of standard unit vectors in R3 and explores whether the vector (0, 4, 5) is contained in the span of vectors (i + j) and k. It provides both algebraic and geometric justifications. Algebraically, it demonstrates that no scalars exist to express (0, 4, 5) as a linear combination of (i + j) and k, as shown by a contradiction in the row-reduced matrix. Geometrically, it illustrates that the vector (0, 4, 5) does not lie in the plane formed by (i + j) and k, confirming it is not in the span.

What are the standard unit vectors in R3?

• a, b, c

• x, y, z

• p, q, r

• i, j, k

What is the problem statement regarding the vector 0, 4, 5?

• Is it orthogonal to i, j, k?

• Is it a unit vector?

• Is it in the span of i, j, k?

• Is it in the span of i + j, k?

What does the span of a set of vectors represent?

• The difference between the vectors

• The product of the vectors

• The sum of the vectors

• The set of all possible linear combinations of the vectors

What must exist for the vector 0, 4, 5 to be in the span of i + j and k?

• Two vectors

• Two scalars

• Two matrices

• Two equations

What is the result of the augmented matrix in reduced row echelon form?

• A solution exists

• No solution exists

• A unique solution exists

• Infinite solutions exist

What does the contradiction in the third row of the matrix indicate?

• The system has a solution

• The system has infinite solutions

• The system is inconsistent

• The system has no solution

Why can't the vector 0, 4, 5 be written as a linear combination of i + j and k?

• Because it is orthogonal to i + j and k

• Because it is a zero vector

• Because the first two components must be the same

• Because it is not a unit vector

What does the geometric justification involve?

• A numerical calculation

• A theoretical explanation

• A matrix representation

• A graph of the vectors

What color is used to represent the vector 0, 4, 5 in the geometric justification?

• Green

• Purple

• Red

• Blue

What does the blue vector's position relative to the plane indicate?

• It is perpendicular to the plane

• It is not in the span

• It is in the span

• It is parallel to the plane