This work considers two related learning problems in a federated attack prone setting: federated principal components analysis (PCA) and federated low rank column-wise sensing (LRCS). The node attacks are assumed to be Byzantine which means that the attackers are omniscient and can collude. We introduce a novel provably Byzantine-resilient communication-efficient and sampleefficient algorithm, called Subspace-Median, that solves the PCA problem and is a key part of the solution for the LRCS problem. We also study the most natural Byzantine-resilient solution for federated PCA, a geometric median based modification of the federated power method, and explain why it is not useful. Our second main contribution is a complete alternating gradient descent (GD) and minimization (altGDmin) algorithm for Byzantine-resilient horizontally federated LRCS and sample and communication complexity guarantees for it. Extensive simulation experiments are used to corroborate our theoretical guarantees. The ideas that we develop for LRCS are easily extendable to other LR recovery problems as well.