linear algebra lec 1
Q: What is a scalar?
A: A single number, often from the real numbers R\mathbb{R}.
Q: What is a vector?
A: An ordered list of numbers, representing magnitude and direction.
Q: What does R3\mathbb{R}^3 represent?
A: The set of all 3-dimensional real vectors.
Q: Give an example of a vector in R2\mathbb{R}^2.
A: [2−1]\begin{bmatrix} 2 \\ -1 \end{bmatrix}
Q: Does vector addition follow the associative property?
A: Yes, ( (\mathbf{a} + \mathbf{b}) +
Q: What is the dimension of [5310]\begin{bmatrix} 5 \\ 3 \\ 1 \\ 0
\end{bmatrix}?
A: 4
Q: Define a zero vector in R3\mathbb{R}^3.
A: [000]\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
Q: What is the length (magnitude) of [34]\begin{bmatrix} 3 \\ 4
\end{bmatrix}?
A: 5
Q: What is a row vector?
A: A vector with components in a single row, like [1,2,3][1, 2, 3]
, Q: What is the geometric representation of a vector?
A: An arrow from the origin to a point in space.
Q: Define vector space informally.
A: A set of vectors that can be added together and scaled.
Q: True or False: All vectors in R3\mathbb{R}^3 have exactly 3
components.
A: True
Q: True or False: Scalar multiplication changes a vector’s direction.
A: False (unless the scalar is negative)
Q: True or False: The zero vector has a magnitude of 0.
A: True
Q: True or False: The vector [00]\begin{bmatrix} 0 \\ 0 \end{bmatrix}
has a defined direction.
A: False
Q: What is a basis? (briefly)
A: A minimal set of linearly independent vectors that span a space.
Q: Are unit vectors always linearly independent?
A: Not necessarily—it depends on their direction and dimension.
Q: Can a set of vectors contain the zero vector and still be linearly
independent?
A: No
Q: If two vectors are orthogonal, are they linearly independent?
A: Yes (in Rn\mathbb{R}^n)
Q: Compute [23]+[1−5]\begin{bmatrix} 2 \\ 3 \end{bmatrix} +
\begin{bmatrix} 1 \\ -5 \end{bmatrix}
A: [3−2]\begin{bmatrix} 3 \\ -2 \end{bmatrix}
Q: What is a scalar?
A: A single number, often from the real numbers R\mathbb{R}.
Q: What is a vector?
A: An ordered list of numbers, representing magnitude and direction.
Q: What does R3\mathbb{R}^3 represent?
A: The set of all 3-dimensional real vectors.
Q: Give an example of a vector in R2\mathbb{R}^2.
A: [2−1]\begin{bmatrix} 2 \\ -1 \end{bmatrix}
Q: Does vector addition follow the associative property?
A: Yes, ( (\mathbf{a} + \mathbf{b}) +
Q: What is the dimension of [5310]\begin{bmatrix} 5 \\ 3 \\ 1 \\ 0
\end{bmatrix}?
A: 4
Q: Define a zero vector in R3\mathbb{R}^3.
A: [000]\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}
Q: What is the length (magnitude) of [34]\begin{bmatrix} 3 \\ 4
\end{bmatrix}?
A: 5
Q: What is a row vector?
A: A vector with components in a single row, like [1,2,3][1, 2, 3]
, Q: What is the geometric representation of a vector?
A: An arrow from the origin to a point in space.
Q: Define vector space informally.
A: A set of vectors that can be added together and scaled.
Q: True or False: All vectors in R3\mathbb{R}^3 have exactly 3
components.
A: True
Q: True or False: Scalar multiplication changes a vector’s direction.
A: False (unless the scalar is negative)
Q: True or False: The zero vector has a magnitude of 0.
A: True
Q: True or False: The vector [00]\begin{bmatrix} 0 \\ 0 \end{bmatrix}
has a defined direction.
A: False
Q: What is a basis? (briefly)
A: A minimal set of linearly independent vectors that span a space.
Q: Are unit vectors always linearly independent?
A: Not necessarily—it depends on their direction and dimension.
Q: Can a set of vectors contain the zero vector and still be linearly
independent?
A: No
Q: If two vectors are orthogonal, are they linearly independent?
A: Yes (in Rn\mathbb{R}^n)
Q: Compute [23]+[1−5]\begin{bmatrix} 2 \\ 3 \end{bmatrix} +
\begin{bmatrix} 1 \\ -5 \end{bmatrix}
A: [3−2]\begin{bmatrix} 3 \\ -2 \end{bmatrix}
