Understanding Linearly Independent Vectors

What is the unique representation of a vector in R2 using standard basis vectors?

• As a difference of I and J

• As a linear combination of I and J

• As a sum of any two vectors

• As a product of I and J

What happens when a third axis is introduced to a two-dimensional space?

• Vectors can only be represented in one way

• Vectors lose their original components

• Vectors can be represented in infinitely many ways

• Vectors become dependent on the third axis

What does it mean for a set of vectors to be linearly independent?

• Vectors can be expressed in multiple ways

• No vector can be expressed as a linear combination of others

• Vectors are dependent on each other

• All vectors can be expressed as a linear combination of others

According to the theorem, when is a set of vectors linearly independent?

• When the sum of vectors equals zero

• When the only solution is the trivial solution

• When vectors can be expressed in terms of each other

• When vectors have non-zero coefficients

What is the trivial solution in the context of linear independence?

• A solution where vectors are dependent

• A solution where vectors are independent

• A solution where all coefficients are one

• A solution where all coefficients are zero

How can Gaussian elimination be used to determine linear independence?

• By finding the sum of all vectors

• By reducing the matrix to the identity matrix

• By expressing vectors as a product

• By adding a new vector to the set

What does a zero determinant indicate about a set of vectors?

• The vectors can be expressed uniquely

• The vectors are linearly independent

• The vectors form a basis

• The vectors are linearly dependent

What is the significance of a non-zero determinant in determining linear independence?

• It shows vectors can be expressed in multiple ways

• It indicates the vectors are linearly dependent

• It indicates the vectors are linearly independent

• It shows vectors are dependent on each other

What is a free variable in the context of Gaussian elimination?

• A variable that is always zero

• A variable that can take any value

• A variable that is fixed

• A variable that is dependent on others

What does it mean if Gaussian elimination does not lead to the identity matrix?

• The vectors cannot be expressed as a combination

• The vectors form a unique basis

• The vectors are linearly dependent

• The vectors are linearly independent