Mr Ng's MA1513 Championship - Round 3

Which of the following are examples of subspaces in ℝ²?

• The set of all vectors lying on the line y=1

• The set of all vectors lying on the y-axis

• The set of all vectors lying on the line y=mx, where m is a non-zero constant

• The set of all vectors with length 1

How can you check for linear independence in a set of vectors?

• Compute their dot products.

• Compute their cross products.

• Check their magnitudes.

• Solve the equation

  c₁v₁+⋯+c_nv_n=0.

Which of the following statements are TRUE?

• A basis is a set of linearly independent vectors that spans the vector space.

• The number of vectors in a basis is always equal to the dimension of the vector space.

• Every vector in the vector space can be expressed as a unique linear combination of the basis vectors.

• The basis vectors of a vector space must be orthogonal to each other.

When the projection of a vector onto a subspace is the zero vector, what can we conclude about the vector?

• The vector lies entirely within the subspace.

• The vector is orthogonal to the subspace.

• The vector is perpendicular to itself.

• The vector has no projection onto any subspace.

What is the aim of the least squares solution?

• To find the best-fitting line that passes through all data points exactly.

• To minimize the sum of absolute errors between observed data points and predicted values.

• To minimize the sum of squared errors between observed data points and predicted values.

• To maximize the sum of squared errors between observed data points and predicted values.