Understanding Linear Subspaces

The video tutorial explains the concept of linear subspaces in Rn, defining them as subsets of vectors that satisfy certain conditions: containing the zero vector, closure under scalar multiplication, and closure under addition. Examples are provided to illustrate both valid subspaces and sets that do not qualify as subspaces. The tutorial also discusses the span of vectors, demonstrating that it forms a valid subspace.

What is the definition of Rn in the context of linear algebra?

• A set of matrices with n rows

• A set of vectors with n components

• A set of equations with n variables

• A set of scalars with n values

Which of the following is NOT a condition for a subset to be a subspace of Rn?

• Contains the zero vector

• Closed under addition

• Closed under scalar multiplication

• Closed under vector subtraction

What happens when you multiply a vector in a subspace by a scalar?

• It becomes the zero vector

• It leaves the subspace

• It becomes a unit vector

• It remains in the subspace

In the context of subspaces, what does 'closure under addition' mean?

• Adding two vectors in the set results in a vector outside the set

• Adding a vector and a matrix results in a vector in the set

• Adding two vectors in the set results in another vector in the set

• Adding a vector and a scalar results in a vector in the set

In the example of a simple subspace in R3, what is the only vector contained in the set?

• The unit vector

• The zero vector

• The identity vector

• A random vector

Why is the subset S in R2, where x1 is greater than or equal to 0, not a subspace?

• It is not closed under scalar multiplication

• It is not closed under vector subtraction

• It is not closed under addition

• It does not contain the zero vector

What does the span of a set of vectors represent?

• The set of all possible linear combinations of those vectors

• The set of all possible scalar multiples of those vectors

• The set of all possible additions of those vectors

• The set of all possible subtractions of those vectors

How can you verify if a span of vectors forms a valid subspace?

• Check if it contains the unit vector and is closed under addition and scalar multiplication

• Check if it contains the zero vector and is closed under subtraction and scalar multiplication

• Check if it contains the unit vector and is closed under subtraction and scalar multiplication

• Check if it contains the zero vector and is closed under addition and scalar multiplication

What is the result of adding two vectors from the span of a single vector?

• A unit vector

• Another vector in the span

• The zero vector

• A vector outside the span

Why is the span of a single vector still considered a subspace?

• It contains the unit vector and is closed under addition and scalar multiplication

• It contains the unit vector and is closed under subtraction and scalar multiplication

• It contains the zero vector and is closed under addition and scalar multiplication

• It contains the zero vector and is closed under subtraction and scalar multiplication